raines bryan dennis
TRANSCRIPT
SYSTEMATIC DESIGN OF MULTIPLE ANTENNASYSTEMS USING CHARACTERISTIC MODES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of the Ohio State University
By
Bryan Dennis Raines, MSEE, BSEE
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2011
Dissertation Committee:
Professor Roberto G. Rojas, Ph.D., Advisor
Professor Robert Garbacz, Ph.D.
Professor Fernando Teixeira, Ph.D.
c© Copyright by
Bryan Dennis Raines
2011
ABSTRACT
Complex multiple antenna systems are emerging today as market pressures to
improve link capacity and reliability on complex integrated platforms. Complexity is
therefore inherent in both the analysis and design of such systems. Modal analysis of
various types has been used in a variety of fields to manage this inherent complexity.
For single antenna systems, the Theory of Characteristic Modes (CM) has been used
to great success in designing single antennas. In this dissertation, four interconnected
topics are discussed, which are crucial to the proper extension of Characteristic Modes
to complex antenna systems, especially multiple antenna systems.
First, existing characteristic mode systems are reviewed and their common prop-
erties examined. A software architecture is then discussed which enables the compu-
tation of characteristic modes in general, requiring only that the particular system
definition satisfies those common properties. The software is used to perform all
computations, analyses, and designs in this dissertation.
Second, a general, high-performance, wideband mode tracking system is proposed.
It is shown to be efficient and robust through several challenging examples. It is the
first robust tracker proposed in connection with Characteristic Modes.
Third, a procedure is proposed to compute the number, location, and even volt-
ages of ports on a given antenna or antennas using a CM description of the prob-
lem. Its mathematical construction is inspired by results from the emerging field of
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Compressed Sensing. Its generality is demonstrated through a number of simulated
designs.
Lastly, two new modal systems related to CM are proposed. One modal sys-
tem, Subsystem Classical Characteristic Modes, produces modes which successively
optimize the ratio of stored power to radiated power for an antenna embedded in
a multistructure system. The other modal system, Target Coupling Characteristic
Modes, produces modes which successively optimize the ratio of induced current in-
tensity on a target structure to the current intensity on a source antenna. The modal
systems are related through a projection matrix, which demonstrates the tradeoff be-
tween mutual coupling and radiation properties for a given multiple antenna system.
The two modal systems are applied to several examples to validate their properties.
Then, they are used in a new design procedure to systematically reduce the mutual
coupling between two parallel dipoles. It is found that through loading an interme-
diate structure suggested by the proposed modal systems, the mutual coupling may
be reduced, also predicted by the modal coupling analysis. The resulting designs
generally improve upon designs available in the literature.
The dissertation concludes with a summary and discussion of future work. The ap-
pendices discusses various definitions, including a novel derivation of Classical Char-
acteristic Modes from the starting point of orthogonal eigenpatterns.
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To Jesus, His Church, Genevieve, Samuel and Lucia.
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ACKNOWLEDGMENTS
I would like to take this opportunity to express my enduring gratitude to my
advisor, Professor Roberto G. Rojas for his guidance, encouragement and patience
during my years at OSU, particularly during this dissertation. His thorough approach
to investigating both theoretical and practical problems has truly set a high standard
that I hope to meet in my career. I would also like to thank Professor Robert G.
Garbacz and Professor Fernando L. Teixeira for their interest, time and energy in
serving on my dissertation committee.
I would also like to thank my friends at the ElectroScience Laboratory (ESL)
for their support during my doctoral studies, especially Khaled, Mutaz, Shadi, Niru,
Wale, Brandan, Renaud, and Keum Su. Our many discussions on electromagnetics
and metaphysics have been thoroughly edifying and served much more than mere
procrastination. My time has been wonderful here substantially because of you.
My special thanks goes out to my former advisor, Professor Dean Arakaki at
Cal Poly, for encouraging me to pursue this crazy course of action into the arena of
graduate electromagnetic engineering.
My love goes out to my parents, Thuy and Rudy and Dennis, for their extraordi-
nary support and encouragement throughout my life in its various phases. My love
also goes out to my new parents-in-law, Katie and Dan, for their prayers, love and
encouragement throughout my graduate studies.
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I sincerely thank my wonderful wife for her love, support, patience, understanding
and a whole host of other adjectives throughout our marriage, especially in connection
with graduate school. Thank you for lifting me to the Lord, for truly you are the
most amazing of all my blessings from Him. I love you, Mrs. Raines. I thank
my darling children for sharing their father so often with others, often beyond their
comprehension. I look forward to our lives in our next stage of life.
Finally, I thank God, creator of both Maxwell’s equations and the universe and
my personal life, a staggering dynamic range. I thank you, Christ, for filling me
everyday with purpose and faith, giving me my marching orders kindly and always
when I needed them. I thank you for the Catholic Church. I thank you for what I
know and what I don’t know, for who I am and who I am not. May I use this degree
in your garden well. I love you.
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VITA
1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. in Electrical Engineering, CaliforniaPolytechnic State University at San LuisObispo
2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. in Electrical and Computer Engi-neering, The Ohio State University
2005-2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dean’s Enrichment Fellow, The Ohio StateUniversity
2006-2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, The OhioState University
2007-2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Northrop Grumman Fellow
2009-2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, The OhioState University
2010-2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dean’s Enrichment Fellow, The Ohio StateUniversity
2011-present . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, The OhioState University
vii
PUBLICATIONS
JOURNALS
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Application of Characteristic Modesand Non-Foster Multiport Loading to the Design of Broadband Antennas,” IEEETrans. Antennas Prop., vol. AP-58, no. 1, pp. 203-207, Jan. 2010
K. A. Obeidat, B. D. Raines, R. G. Rojas, and B. T. Strojny, “Design of FrequencyReconfigurable Antennas Using the Theory of Characteristic Modes,” IEEE Trans.Antennas Prop., vol. AP-58, no. 10, pp. 3106-3113, Oct. 2010
K. A. Obeidat, B. D. Raines, and R. G. Rojas. “Discussion of series and parallelresonance phenomena in the input impedance of antennas,” Radio Science, 45, Dec.2010
N. K. Nahar, B. D. Raines, B. T. Strojny, and R. G. Rojas, “Wideband Antenna ArrayBeam Steering with Free-Space Optical True-Time Delay Engine,” IET Microwaves,Antennas and Prop., vol. 5, no. 6, pp. 740-746, May 2011
CONFERENCES
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Antenna design and analysis usingcharacteristic modes,” IEEE Antennas and Propagation Symposium, pp. 59935996,June 2007
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Broadband antenna synthesis usingcharacteristic modes,” URSI North American Radio Science Meeting, 2007
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Design of omnidirectional electricallysmall Vee- shaped antenna using characteristic modes,” URSI 2008 National RadioScience Meeting, January 2008
B. D. Raines, K. A. Obeidat, and R. G. Rojas, “Characteristic mode-based design andanalysis of an electrically small planar spiral antenna with omnidirectional pattern,”IEEE Antennas and Propagation Symposium, July 2008
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Design and analysis of a helicalspherical antenna using the Theory of Characteristic Modes,” IEEE Antennas andPropagation Symposium, July 2008
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K. A. Obeidat, B. D. Raines, and R. G. Rojas “Design of Antenna Conformal to V-shaped Tail of UAV Based On the Method of Characteristic Modes,” EWCA Berlin,Germany, March 23-27, 2009
N. K. Nahar, B. D. Raines, B. T. Strojny, and R. G. Rojas, “A Practical Approachto Beam Steering with White Cell True-Time Delay Engine,” IEEE Antennas andPropagation Symposium Conference, Charleston, SC, June 2009
K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Analysis of Antenna Input ImpedanceResonances in Terms of Characteristic Modes,” 2009 USNC/URSI National RadioScience Meeting, North Charleston, SC, June 2009
B. D. Raines, and R. G. Rojas, “Design of Multiband Reconfigurable Antennas,”EuCAP 2010, Barcelona, Spain, April 2010
B. D. Raines, and R. G. Rojas, “Wideband Characteristic Mode Tracking,” EuCAP2011, Rome, Italy, April 2011
B. D. Raines, B. T. Strojny, and R. G. Rojas, “Systematic Characteristic Mode AidedFeed Design,” USNC/URSI National Radio Science Meeting, June 2011
E. A. Elghannai, B. D. Raines, and R. G. Rojas, “Design of Electrically Small An-tennas with Multiport Non-Foster Loading using Network Characteristic Modes,”USNC/URSI National Radio Science Meeting, June 2011
PATENTS
R. G. Rojas, B. D. Raines, and K. A. Obeidat, “Implementation of Ultra-Wide Band(UWB) Electrically Small Antennas by Means of Distributed Non-Foster Loading,”U.S. Patent 7,898,493
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FIELDS OF STUDY
Major Field: Electrical and Computer Engineering
Minor Field: VLSI
Specialization: Electromagnetics
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TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
CHAPTER PAGE
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Engineering Complex Systems . . . . . . . . . . . . . . . . 21.1.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Characteristic Mode Analysis . . . . . . . . . . . . . . . . 41.1.4 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 Key Contributions . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Characteristic Modes: 1960-1980 . . . . . . . . . . . . . . . 61.2.2 Characteristic Modes: Recent Work . . . . . . . . . . . . . 111.2.3 Mutual Coupling . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . 15
2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Generalized Rayleigh Quotient . . . . . . . . . . . . . . . . 182.1.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . 192.1.4 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . 222.1.5 Conceptual Implications . . . . . . . . . . . . . . . . . . . 23
2.2 Characteristic Mode Theory . . . . . . . . . . . . . . . . . . . . . 232.2.1 Classical CM . . . . . . . . . . . . . . . . . . . . . . . . . 24
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2.2.2 Inagaki CM . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.3 Generalized CM . . . . . . . . . . . . . . . . . . . . . . . . 272.2.4 General Requirements of Modal Systems . . . . . . . . . . 29
2.3 Outline of Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 UCM Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Computing the Generalized Eigenvalue Problem . . . . . . 342.4.3 Computing Modal Weighting Coefficients . . . . . . . . . . 352.4.4 Performance Considerations . . . . . . . . . . . . . . . . . 35
3 Wideband Characteristic Mode Tracking . . . . . . . . . . . . . . . . . 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Modal Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . 393.2.2 Proposed Tracking Algorithm . . . . . . . . . . . . . . . . 42
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 Linear Wire Dipole . . . . . . . . . . . . . . . . . . . . . . 473.3.2 Spherical Helix . . . . . . . . . . . . . . . . . . . . . . . . 483.3.3 Rectangular Microstrip Patch . . . . . . . . . . . . . . . . 49
3.4 Alternative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 523.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Computer-Aided Antenna Feed Design Using Characteristic Modes . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . 604.3.2 Iterations Required . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Example: Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . 654.4.1 Design 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.2 Design 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4.3 Design 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Example: Folded Spherical Helix Antenna . . . . . . . . . . . . . . 734.5.1 Design 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.2 Design 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Systematic Mutual Coupling Reduction Through Modal Analysis . . . 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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5.2.1 Partitioned Matrices: Two Structures . . . . . . . . . . . . 885.2.2 Partitioned Matrices: Three Structures . . . . . . . . . . . 905.2.3 Proposed Radiation Mode System . . . . . . . . . . . . . . 925.2.4 Proposed Coupling Mode System . . . . . . . . . . . . . . 935.2.5 Expressing Modes in Different Modal System . . . . . . . . 95
5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.1 Example: Dipole Pair . . . . . . . . . . . . . . . . . . . . . 995.3.2 Example: Yagi-Uda . . . . . . . . . . . . . . . . . . . . . . 1065.3.3 Example: Dipole Beside Microstrip Patch . . . . . . . . . . 114
5.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.4.1 Original Design . . . . . . . . . . . . . . . . . . . . . . . . 1225.4.2 Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4.3 Design A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.4 Design B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.4.5 Design C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.4.6 Comparison of Designs . . . . . . . . . . . . . . . . . . . . 137
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1 Summary and Conclusions of the Dissertation . . . . . . . . . . . . 1426.1.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.1.2 Wideband Mode Tracker . . . . . . . . . . . . . . . . . . . 1436.1.3 Antenna Feed Design . . . . . . . . . . . . . . . . . . . . . 1446.1.4 Mutual Coupling Reduction . . . . . . . . . . . . . . . . . 144
6.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . 1466.2.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.2.2 Wideband Mode Tracking . . . . . . . . . . . . . . . . . . 1476.2.3 Antenna Feed Design . . . . . . . . . . . . . . . . . . . . . 1486.2.4 Mutual Coupling Reduction . . . . . . . . . . . . . . . . . 149
APPENDICES
A Alternative Derivation for Classical Characteristic Modes . . . . . . . . 151
A.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B Modal Input Power in Classical Characteristic Modes . . . . . . . . . . 156
B.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
C Modal Weighting Coefficient Derivation for General Modal Systems . . 158
C.1 General Modal Systems . . . . . . . . . . . . . . . . . . . . . . . . 158C.2 Classical CM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.2.1 Alternative Definition . . . . . . . . . . . . . . . . . . . . . 160
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C.3 Generalized CM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161C.4 Other CM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
D Modal Weighting Coefficient Derivation Using Modal Patterns . . . . . 163
D.1 Derivation of Modal Weights . . . . . . . . . . . . . . . . . . . . . 163D.2 Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . 164
E Mutual Impedance Derivation . . . . . . . . . . . . . . . . . . . . . . . 167
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
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LIST OF FIGURES
FIGURE PAGE
1.1 Desired and realized modal far-field patterns . . . . . . . . . . . . . . 10
1.2 Synthesized characteristic surface at 100 MHz . . . . . . . . . . . . . 10
2.1 UCM analysis software architecture . . . . . . . . . . . . . . . . . . . 31
3.1 Dipole eigenvalue spectrum without modal tracking . . . . . . . . . . 41
3.2 1.2 meter dipole eigenvalue spectrum with 32 tracked modes . . . . . 48
3.3 Spherical helix antenna geometry . . . . . . . . . . . . . . . . . . . . 49
3.4 Spherical helix antenna eigenvalue spectrum with 40 tracked modes . 50
3.5 Microstrip patch antenna geometry . . . . . . . . . . . . . . . . . . . 50
3.6 Patch antenna eigenvalue spectrum with 10 tracked modes . . . . . . 51
4.1 Initial port voltage magnitudes versus frequency . . . . . . . . . . . . 62
4.2 Averaged port voltages . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 1.2 meter dipole characteristic mode eigenvalue spectrum . . . . . . . 65
4.4 Dipole design 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Dipole design 1 realized modal weight magnitudes . . . . . . . . . . . 67
4.6 Dipole design 1 directive elevation patterns . . . . . . . . . . . . . . 67
4.7 Dipole design 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.8 Dipole design 2 realized modal weight magnitudes . . . . . . . . . . . 70
4.9 Dipole design 2 directive elevation patterns . . . . . . . . . . . . . . 70
4.10 Dipole design 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
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4.11 Dipole design 3 realized modal weight magnitudes . . . . . . . . . . . 72
4.12 Dipole design 3 directive elevation patterns . . . . . . . . . . . . . . 72
4.13 Four-arm folded spherical helix antenna geometry from [1] . . . . . . 73
4.14 Four-arm folded spherical helix antenna eigenvalue spectrum . . . . . 74
4.15 Folded spherical helix input reflectance versus frequency . . . . . . . 75
4.16 Folded spherical helix impedance Q versus frequency . . . . . . . . . 75
4.17 Folded spherical helix directive elevation patterns . . . . . . . . . . . 76
4.18 Desired omnidirectional pattern . . . . . . . . . . . . . . . . . . . . . 76
4.19 Folded spherical helix design 1 desired modal weight magnitudes . . . 77
4.20 Folded spherical helix port voltages for solution iteration 1 . . . . . . 78
4.21 Folded spherical helix port voltages for solution iteration 3 . . . . . . 78
4.22 Folded spherical design 1 . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.23 Folded spherical helix design 1 realized modal weight magnitudes . . 79
4.24 Folded spherical helix design 1 total elevation gain patterns . . . . . 80
4.25 Folded spherical helix design 2 (with axis) . . . . . . . . . . . . . . . 81
4.26 Folded spherical helix design 2 input reflectance versus frequency . . 81
4.27 Folded spherical helix design 2 impedance Q versus frequency . . . . 82
4.28 Folded spherical helix design 2 total elevation gain patterns . . . . . 82
5.1 Example of desirable and undesirable radiation modes according tosubsystem classical characteristic modes . . . . . . . . . . . . . . . . 97
5.2 Example of desirable and undesirable coupling modes according totarget coupling characteristic modes . . . . . . . . . . . . . . . . . . 97
5.3 Example of the projection of the coupling modes onto the radiationmodes at 2.4 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Geometry for the Dipole Pair analysis example . . . . . . . . . . . . 99
5.5 S11 and S12 of the dipole pair . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Classical characteristic mode eigenvalue spectrum of a single dipole . 101
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5.7 Classical characteristic mode modal weighting coefficients of a singledipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.8 Classical characteristic mode eigenvalue spectrum of system composedof two dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.9 Radiation mode eigenvalue spectrum of Dipole 1 in the presence ofDipole 2, with port 2 terminated with 50Ω . . . . . . . . . . . . . . . 103
5.10 Modal weighting coefficients of radiation modes of Dipole 1 . . . . . 103
5.11 Coupling mode eigenvalue spectrum of Dipole 1 (source antenna) cou-pling to the port of Dipole 2 (target structure) . . . . . . . . . . . . 104
5.12 Modal weighting coefficients of coupling modes . . . . . . . . . . . . 105
5.13 Yagi Antenna 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.14 Yagi Antenna 1 radiation mode eigenvalue spectrum . . . . . . . . . 107
5.15 Yagi Antenna 1 radiation modal weighting coefficients . . . . . . . . 108
5.16 Yagi Antenna 1 coupling mode eigenvalue spectrum . . . . . . . . . . 108
5.17 Yagi Antenna 1 coupling modal weighting coefficients . . . . . . . . . 109
5.18 Yagi Antenna 1 coupling mode to radiation mode projection matrix . 110
5.19 Yagi Antenna 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.20 Yagi Antenna 2 radiation mode eigenvalue spectrum . . . . . . . . . 111
5.21 Yagi Antenna 2 radiation modal weighting coefficients . . . . . . . . 112
5.22 Yagi Antenna 2 coupling mode eigenvalue spectrum . . . . . . . . . . 112
5.23 Yagi Antenna 2 coupling modal weighting coefficients . . . . . . . . . 113
5.24 Dipole beside microstrip patch geometry . . . . . . . . . . . . . . . . 114
5.25 Input reflectance of dipole and patch antenna system . . . . . . . . . 115
5.26 Mutual coupling between dipole and patch antenna . . . . . . . . . . 115
5.27 Dipole-patch radiation mode eigenvalue spectrum . . . . . . . . . . . 116
5.28 Dipole-patch radiation modal weighting coefficients . . . . . . . . . . 117
5.29 Dipole-patch coupling mode eigenvalue spectrum . . . . . . . . . . . 118
5.30 Dipole-patch coupling modal weighting coefficients . . . . . . . . . . 118
xvii
5.31 Dipole-patch coupling mode to radiation mode projection matrix . . 119
5.32 Dipole 1 coupling mode to radiation mode projection matrix . . . . . 121
5.33 Dipole pair with parasitic structure geometry . . . . . . . . . . . . . 123
5.34 Dipole 1 radiation mode eigenvalue spectrum with the original couplerdesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.35 Dipole 1 radiation modal weighting coefficients with the original cou-pler design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.36 Original coupler radiation mode eigenvalue spectrum . . . . . . . . . 125
5.37 Original coupler design input reflectance . . . . . . . . . . . . . . . . 126
5.38 Original coupler design mutual coupling . . . . . . . . . . . . . . . . 126
5.39 How the original coupler design induces a mixture of Mode 1 and Mode3 on Dipole 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.40 Design A ideal loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.41 Coupler Design A input reflectance . . . . . . . . . . . . . . . . . . . 130
5.42 Coupler Design A mutual coupling . . . . . . . . . . . . . . . . . . . 130
5.43 Design A source dipole radiation mode eigenvalue spectrum . . . . . 131
5.44 Design A source dipole radiation modal weighting coefficients . . . . 132
5.45 Design B ideal loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.46 Coupler Design B input reflectance . . . . . . . . . . . . . . . . . . . 133
5.47 Coupler Design B mutual coupling . . . . . . . . . . . . . . . . . . . 134
5.48 Design C ideal loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.49 Coupler Design C input reflectance . . . . . . . . . . . . . . . . . . . 135
5.50 Coupler Design C mutual coupling . . . . . . . . . . . . . . . . . . . 136
5.51 Comparison of input reflectance at Port 1 for all designs . . . . . . . 137
5.52 Comparison of mutual coupling for all designs . . . . . . . . . . . . . 138
5.53 Comparison of radiation efficiency for all designs (ideal loads) . . . . 139
5.54 Comparison of embedded vertical gains at 2.45 GHz at θ = 0 (az-imuthal cuts) for all designs . . . . . . . . . . . . . . . . . . . . . . . 140
xviii
E.1 Two problems for mutual impedance . . . . . . . . . . . . . . . . . . 168
xix
CHAPTER 1
INTRODUCTION
1.1 Background and Motivation
Antenna engineering, like other modern engineering disciplines, involves the appli-
cation of physical analysis techniques to solving design problems. For a particular
antenna design problem, such as designing a simple linear dipole antenna to give
a certain radiation pattern and peak gain level over a narrow frequency band, there
may be several competing design techniques. One typical approach is to apply specific
knowledge of the given antenna geometry parameters (in this case, the dipole length
and feed location) determined from experience and antenna theory [2] to achieve the
desired result. Usually, this specific knowledge is physical in nature and is derived
from an approximate physical and analytical model that is properly parameterized.
Deriving this accurate physical, if not entirely analytical, model for an arbitrary an-
tenna is extremely difficult, as evidenced by the lack of literature on this subject.
Reporting on antenna design techniques has therefore evolved into reporting on spe-
cific antennas with specific parameterizations. A notable exception has been the case
of frequency-independent antennas [3, 2], which give general principles for the class
of frequency-independent antennas, such as spiral antennas [4], conical antennas [5],
and log-periodic antennas [6].
While the literature extensively catalogs existing antenna systems optimized for
1
various design problems and is occasionally peppered with general antenna design
principles [3, 7], it still lacks a design methodology applicable to arbitrary antennas
(or at least, an extremely large class of antennas). To attempt to fill this gap is a
grand challenge for antenna designers going forward, as conformal and/or integrated
antenna design is becoming a significant trend in the industry [8, 9, 10]. To provide
a reasonably general design methodology, we first need to have a general analysis
methodology.
1.1.1 Engineering Complex Systems
Engineering a system requires a systematic approach, especially in analysis. Analyz-
ing a complex system requires one to decompose the system into simpler components.
By simpler, we do not necessarily require the component to be simple to make or even
necessarily simple in overall function. Rather, since the overall analysis is directed
toward a particular set of engineering goals, we only require the system components
to be simple in the context of meeting the overall problem.
For example, a rocket is an enormously complex system. Similarly, a satellite
is a complex system. If the engineering problem is to supply television signals to
wide geographical areas, one solution is to launch a satellite equipped with television
transmission equipment using a rocket. Functionally, breaking the problem (TV
signals over large geographical areas) down into three components (rocket, satellite
with transceiver capability, and stationary planetary orbit) makes the problem simpler
precisely because the role of each component is unique and well-understood. The
complexity of each component may be similarly decomposed into a set of functionally
simpler elements until each element’s role is so well understood in the context of the
overall problem that it may be engineered. Again, the final engineered components
2
may be complex overall, but its functionally simple role in the overall problem context
allowed it to be well-engineered.
1.1.2 Modal Analysis
Managing complexity using modal analysis is a popular technique. It also happens
to be quite general. In the field of electromagnetic engineering, it is common practice
to engineer waveguides using modal analysis, which generates excitation-independent
modal fields based solely upon the cross-section of the proposed waveguide [11, 12].
Another popular analysis experimental electromagnetic modal analysis technique for
time domain problems is the Singularity Expansion Method (SEM) [13]. It will not
be examined in further detail here because of this work’s focus on frequency-domain
methods.
In the fields of aerospace and mechanical engineering, mechanical modal analy-
sis (known simply as modal analysis) has revolutionized the design of components
[14]. Using the Finite Element Method (FEM) [15], mechanical modal analysis also
generates excitation-independent modal deflections. These modes may be computed
from FEM or measured experimentally. From this data, structures are modified to be
more earthquake resilient [16, 17], bridges more stable in the presence of wind [18],
and aircraft control surfaces more reliable [19].
In all these cases, the modal analysis allows the problem to be decomposed into
functionally simpler (i.e. unique roles) modes. The modes are excited by some
external forcing function (e.g., a coaxial probe or a gust of wind) and their collective
response determines the overall system’s behavior. Most importantly, a useful modal
description of a system will involve a relatively small number of modes with significant
coupling to the external forcing function. That is, a useful modal system definition
is one where the overall behavior of the system may be well approximated by the
3
excitation of only a few modes. If instead of approximating the system’s behavior
using only a few modes, many modes are required, then it is arguable as to whether
the original problem’s complexity has been satisfactorily managed, if not expanded.
1.1.3 Characteristic Mode Analysis
The Theory of Characteristic Modes (CM), also known as the Theory of Classical
Characteristic Modes (CCM), is a frequency-domain modal analysis system defined
for a large class of electromagnetic radiating and scattering systems, such as antennas.
It was first proposed by Robert Garbacz [20] in 1965 and later formally associated
with the Method of Moments (MoM) [21] by Roger Harrington and Joseph Mautz
[22] in 1971. It is an extremely general modal analysis technique, owing its practical
generality to MoM. It is defined by a generalized eigenvalue problem parameterized
by frequency. Assuming that it is applied to lossless structures [23], CCM provides
for each mode: an eigenvalue indicating the ratio of stored power to radiated power of
that mode at each frequency; orthogonal eigencurrents, or a surface current density
associated with that particular mode; and orthogonal eigenpatterns, or the radiated
far-fields produced by an eigencurrent. Because of the orthogonality, both the total
surface current density and the total far-field pattern may be decomposed into a
weighted summation of modes.
Decomposing the total surface current density or total far-field patten into a
weighted summation of modes is important, not only because each mode has an
eigenvalue describing its suitability as a wideband radiator, but also because the
weights allow one to approximate the total current density or far-field pattern using
a subset of the modes. Usually, the subset is small because the number of modes
with relatively large modal weights is also small. This feature is important because it
allows for general problem complexity to be managed. Most significantly, the modal
4
weights describe how well a particular set of antenna feed ports or incident field couple
power to each mode, giving the designer both quantitative and qualitative insight into
the antenna or scatterer’s physics.
1.1.4 Goals
The overall goal of this work is to provide a systematic approach to designing com-
plex radiator systems, especially those involving more than one antenna. I draw on
my experience in designing actual antenna systems for sponsors, usually involving
coexistence issues, but sometimes involving direct multi-antenna interaction issues.
To solve these problems, I will leverage the benefits of modal analysis to create a new
form of characteristic mode analysis for multiple antenna systems, capable of analyz-
ing coexistence and coupling issues. Necessarily, I will need to make a few related
contributions to making the analysis of single antennas analyzed and designed using
characteristic modes (and related modal systems) treat bandwidth more explicitly.
The organization of the dissertation follows from these general elements.
1.1.5 Key Contributions
In this dissertation, I have made the following key contributions:
• Wideband Mode Tracking: Introduced the first high performance wideband
mode tracking system in the field of characteristic modes
• Computer Aided Feed Design: Introduced the first system to automatically de-
termine the number and placement of feed ports on antennas given a character-
istic mode-based description of the antenna’s operation over a given bandwidth,
leveraging concepts from Compressed Sensing
5
• Radiation Modes: Introduced a new type of characteristic mode analysis suited
to analyzing the radiation characteristics of a single antenna operating in the
presence of other antennas
• Coupling Modes: Introduced a new type of characteristic mode analysis suited
to analyzing the coupling between a single antenna and multiple target struc-
tures in its vicinity
• Designs to Minimize Mutual Coupling Between Two Antennas: Developed de-
signs to minimize the mutual coupling between two closely spaced co-polarized
dipoles using information derived from the radiation modes and coupling modes
1.2 Literature Review
CM theory has been used in the past usually for antenna analysis, and antenna
placement on larger support structures, although there are a few examples of using
CM for antenna design in the literature. This section takes a historical overview of
the literature discussing characteristic modes.
1.2.1 Characteristic Modes: 1960-1980
The theory of Characteristic Modes was first considered by Robert Garbacz in his
dissertation [20] and later summarized in [24].1 It came from considering the problem
1I choose to provide a fuller summary of his work here because it is often glossed over as importantonly in connection with Harrington’s later work. I speculate that Garbacz’s work, in particularthe physical reasoning leading up to the modal method, could enable experimental measurementsof characteristic modes using the scattering or perturbation matrix.
6
of finding scattering eigenfunctions for arbitrary configurations of conducting surfaces.
That is, given a scatterer, a scattering operator S may be formed:
~Eout = S( ~Ein)
where the total far-field ~E is the superposition of the incoming field ~Ein and the
outgoing field ~Eout: ~E = ~Ein + ~Eout.
An associated perturbation operator P may be defined which maps the incoming
field ~Ein to the scattered field ~Es ≡ ~E − ~Ein:
~Es = P ( ~Ein)
Critically, the perturbation operator is normal when the scatterer is lossless,
thereby admitting orthogonal eigenfunctions; Garbacz sought these orthogonal eigen-
functions of the perturbation operator. By construction, these modal fields are such
that if they are incident upon the scatterer, the scattered field is simply a complex
scaled version of themselves.2 He then proposed a way to numerically compute the
modes using a form of MoM with point testing and entire domain basis functions
(i.e., Fourier basis) through an ordinary eigenvalue problem:
[X(α)]T [X(α)]Jn = εn(α)δ
where [X(α)] is the imaginary part of [Z(α)] = [Z]e−jα, α is some real constant, and
δ is the imaginary part of E, the tangential component of the scattered electric field
on the surface of the scatterer.
His technique yielded real modal currents and ensured that each associated modal
tangential electrical field on the scatterer surface was equiphase with its corresponding
2Harrington [22] notes that the modal scattered field is the complex conjugate of the incidentmodal field
7
modal current. Garbacz theorized that for electrically small to intermediate struc-
tures, the number of modes required to compute the scatterer’s response would be
small, justified by several examples involving thin wire structures. For the first time,
a general technique was available to compute and analyze the modal response of an
arbitrary scatterer.
Harrington and Mautz [22, 23] chose to define the characteristic modes of an ar-
bitrary lossless metallic structure through a generalized eigenvalue problem. It is
discussed in much more detail in Chapter 2. Like Garbacz’s technique, it sought
real modal currents whose associated incident or impressed modal electric fields tan-
gential to the surface of the lossless scatterer are equiphase. Instead of defining
the modes to directly minimize the phase variation of the tangential modal electric
field, a generalized eigenvalue problem (GEP) was defined to generate modal cur-
rents such that the associated tangential modal electric fields would be equiphase:
[Z]Jn = (1 + jλn)[R]Jn. The GEP also defined orthogonal modal far-fields. Their
technique required the complex MoM generalized impedance matrix [Z] to be sym-
metric, implying that it should be constructed using the Galerkin method and sub-
domain basis/testing functions. Since the GEP directly used the impedance matrix
from a particular formulation of MoM to produce modal currents and modal far-fields
with the same properties as Garbacz’s technique, it became the de-facto method of
constructing the characteristic modes.
Harrington and others went on to apply CM theory to reducing the radar cross
section of obstacles using reactive loading [25], pattern control using loading [26],
and defining the modes for multiport antennas (network characteristic modes) [27].
He went on to extend characteristic modes (with some limitations) to dielectric and
magnetic bodies [28, 29]. A separate work extended CM theory to use the MFIE
instead of the EFIE MoM formulation to improve its lower frequency behavior [30]
8
for certain scatterers. Important for antennas, the modal self-admittance and mutual
admittance were defined by Garbacz in [31]. Finally, characteristic modes were shown
to be useful for locating a small antenna on a much larger conducting body (airframe)
in a systematic fashion [32].
Dramatically, a pair of papers using properties of characteristic modes (in par-
ticular, Garbacz’s insight that the modal electric field tangential to the surface of a
lossless structure is equiphase) emerged [33, 34], enabling the computation of 3D an-
tenna structures designed to have a dominant mode which radiated a desired pattern.
The structures were limited to bodies of revolution. The key idea of the papers was
to define a far-to-near field transform using spherical modes and then find equiphase
electric field surfaces on which the equivalent equiphase electric current lead by a
specified phase. As an example of its generality, the vertically polarized desired and
realized far-field patterns are shown in Figure 1.1 and the synthesized characteristic
surface (rotationally symmetric about the Z-axis) is shown in Figure 1.2.
Unfortunately, the problem was left open-ended, as it would be difficult to locate
appropriate feed points to excite the synthesized surface such that the desired mode
was dominant. Still, it could be used to generate rotationally-symmetric scatterers
with a specific pattern. More importantly, the concept is not theoretically limited
to rotationally-symmetric scatterers, although the computers of the day practically
limited their investigation to such scatterers.
Garbacz and Inagaki went on to define a new modal method [35, 36], commonly
termed Inagaki characteristic modes. It bore some similarities to Garbacz’s original
approach, but used a GEP to optimize the electric field over some target surface to be
optimized with respect to the source field. The modal currents were still orthogonal
because both matrices in the GEP were Hermitian, but they were complex.
A simplification of Inagaki modes in [37, 38], called generalized characteristic
9
Figure 1.1: Desired and realized modal far-field patterns
Figure 1.2: Synthesized characteristic surface at 100 MHz
10
modes, restored some of the benefits of the original characteristic modes, now termed
classical characteristic modes, while preserving some of the flexibility of the original
Inagaki formulation. Here, the [R] matrix was replaced by another matrix defin-
ing regions of far-field orthogonality. The regions were specified by appropriately
weighting the field points on the far-field sphere. It could be related back to the clas-
sical characteristic modes using a weight of 1 at all far-field points (thus, the term
”generalized”). Generalized CM were later extended by Ethier, et al. [39] to allow
per-polarization far-field weights to be defined.
Despite these advances, characteristic mode analysis did not catch on in the com-
munity (assuming that the literature is an accurate measure of community popular-
ity), probably because MoM could not be used to accurately analyze structures of
practical interest, considering the limitations of computers at the time. Mechanical
modal analysis, on the other hand, did not suffer from the same problems because
of the less demanding requirements of FEM, becoming extremely popular in the me-
chanical, civil, and related engineering communities.
1.2.2 Characteristic Modes: Recent Work
More recently, a few groups worldwide have taken up characteristic mode theory once
again.
CM Theory
There was work defining a special class of characteristic modes to aperture prob-
lems [40] and discussing the particular advantages of using characteristic modes over
standard MoM [41], [42], [43], [44].
Another group in Italy was using characteristic modes in a somewhat different way.
Instead of being used primarily as an analysis tool, they used the modal currents as
11
entire-domain basis functions to accelerate the MoM solution of array problems [45],
[46], [47], made especially clear in their later papers [48], [49]. In [46], they also
provided analytical characteristic modes for finite cylinders in order to verify the
numerical accuracy of their mode computations.
In the same spirit, a separate group published a technique of rapidly computing
RCS using a combination of entire-domain basis functions (i.e. dominant character-
istic mode currents) and AWE (asymptotic waveform evaluation) [50].
Their last major contribution was to apply a recent mathematical result to CM,
enhancing the accuracy of the GEP computation in classical CM [51]. The basic
problem is that the although the [R] matrix in the GEP computation is theoretically
positive-definite for lossless structures without internal resonances, the computed [R]
matrix is sometimes indefinite for numerical reasons [23]. Distinct from the approach
in [23], Angiulli et al. used the Higham-Cheng theorem [52] to define a new pair of
matrices [X ′] and [R′] which are a ”positive-definite pair.” The modes are computed
from the new matrices and transformed back to [X] with a modified positive-definite
[R].
Finally, there was a contribution of a group in Spain to explain the slow con-
vergence in input susceptance [53] observed by Garbacz earlier [31] (i.e. many more
modes were required to reconstruct the input admittance for a given antenna com-
pared to its far-field pattern). They proposed to accelerate convergence through the
introduction of a physically-derived ”source mode,” although it is hypothesized that
the susceptance should in fact contain the contributions of several excited, but poorly
radiating modes, similar to the concept of evanescent modes in guided waves [11].
12
CM in Antenna Analysis
Continuing the line of research started by Newman in using characteristic modes for
antenna placement [32], Strohschein et al. published a dissertation [54] and a con-
ference paper [55]. Especially interesting is the description of how the overall system
modes evolve as the ”probe” antenna is placed at various points on the structure.
Significant research has also been conducted into specifically using characteristic
modes to understand the qualitative, as well as quantitative, behavior of various
antennas by the Spanish since 2000 [56], [57], although one paper was published
earlier in 1989 discussing the analysis of log-periodic antennas using characteristic
modes [58].
Separately, two papers have been published by a Canadian group in 2010 dis-
cussing new computationally simpler metrics for complex antenna analysis [59] and
optimization [60].
Lastly, an analysis by Obeidat and Raines on the modal behavior around series
and parallel resonance for single port antennas was recently published [61]. It’s pri-
mary contribution is the proof that a series resonance is caused primarily by one
characteristic mode, while a parallel resonance is caused by the interaction of multi-
ple modes. This analysis enabled Prof. Rojas’s group to experimentally determine
whether an antenna prototype’s structure was sufficiently similar to simulation by
noting the series resonance frequencies, since the series resonance frequencies are less
sensitive to the feed position than parallel resonances.
CM in Antenna Design
Research into using characteristic modes as an aid to design conformal or otherwise
non-traditional antennas was first conducted by K. P. Murray [62], [63], [64], [65].
Various design metrics were later proposed by the Spanish group to identify modes
13
with broader bandwidth impedance and pattern behavior, which made the design
process of several commercially-popular antennas more controlled [66], [57, 67, 68,
69, 70]. In [71], they summarized much of their work.
Prof. Roberto Rojas and his students at The Ohio State University have been con-
tributing various design procedures for arbitrary lossless antennas, including lumped
reactive loading in order to expand the bandwidth of a single equiphase modal current
[72], [73], [74], lumped reactive loading to realize frequency-reconfigurable antennas
[75], [76], and techniques for improving the bandwidth and gain of electrically small
conformal antennas [77], [78], [8], [79]. Much of the work is summarized in [80].
Another group in Canada has also begun to investigate the application of char-
acteristic modes to antenna design, capitalizing on the orthogonality of modes over
a single antenna to improve port isolation in MIMO antenna design [81] and de-
veloping an extension to generalized characteristic modes to improve the directional
performance of MIMO antennas [39].
1.2.3 Mutual Coupling
The literature concerning mutual coupling and its effects on system performance is
split into two main branches. The first documents the mutual coupling between vari-
ous antennas, ranging from general analysis on two element arrays [82] to the coupling
between two adjacent microstrip patch antennas [83, 84]. The second documents how
to compensate for the presence of mutual coupling in arrays, mainly through signal
processing [85, 86]. Since this work examines a method to reduce mutual coupling
physically, the relevant literature on mutual coupling should concern physical modi-
fications to antennas rather than signal processing algorithms. While certain success
has been achieved with isolating two antennas using a large plate [87], more general
methods have appeared, all of which claim to use resonant parasitic structures to
14
physically reduce mutual coupling [88], [89], [90]. The most general of the methods is
likely the one presented in [90] and I will be examining it in greater detail in Chapter
5. Separately, there has been very promising work to design custom metamaterial
structures for mutual coupling reduction [91].
1.3 Organization of the Dissertation
The dissertation is organized in the following manner. First, Chapter 2 discusses
the necessary background and design methodology supporting the systematic design
of multiple antenna systems using a new form of characteristic modes. The chapter
opens with the mathematical background behind all the forms of characteristic mode
analysis. It then reviews the formulations of each type of characteristic mode analysis.
The chapter concludes with considerations of general modal systems in antenna (and
multiple antenna) systematic design.
Chapter 3 discusses a novel algorithm enabling wideband characteristic mode
tracking. This algorithm is required for any serious systematic modal design of a
wideband antenna system, since each type of characteristic mode analysis defines
a frequency-dependent (generalized) eigenvalue problem. The modes computed at
one frequency must be automatically related to the modes at another frequency in
order for the information to be useful. To the best of my knowledge, it is the first
modal tracking method presented in connection with characteristic modes, although
similarities to tracking in other fields are discussed. In particular, the method can
track a large number of modes using the modal eigenvectors from a generalized eigen-
value problem parameterized by frequency. The method and a naive alternative are
detailed following a brief overview of the relevant elements of CM theory. The two
techniques are applied to three distinct geometries and the results discussed. Relative
15
to alternative tracking techniques from other disciplines implemented in MATLAB
[92], the proposed method features greatly improved performance.
Chapter 4 discusses another novel algorithm enabling the systematic determina-
tion of the number, location, and excitation voltages of feed ports for an arbitrary
lossless antenna given the desired complex modal weights over some frequency range.
The method defines an underdetermined system of equations and draws upon recent
results from the field of Compressed Sensing as well as image processing to obtain a
physically useful solution. Several examples are shown to demonstrate the method’s
capabilities and generality.
Chapter 5 uses the methods from the previous chapters combined with a new
modal system definition specific to multiple antenna systems to explicitly design
such systems with reduced mutual coupling. The modal system is explained through
physical arguments and defined mathematically. The multiple antenna design method
is applied to a practical example and compared with existing work in the literature
to illustrate its use and benefits.
Finally, Chapter 6 summarizes the key contributions of this dissertation and sug-
gestions for future work are given.
16
CHAPTER 2
METHODOLOGY
This chapter will first review some relevant mathematical tools, which shall hopefully
prove useful in understanding the machinery of successful modal systems for radiat-
ing devices. Then, the major types of Characteristic Mode theory will be reviewed
and some generalizations made. Then, we shall present some elements and challenges
unique to multiple antenna systems, summarizing the approach taken in this disser-
tation. Finally, the chapter is rounded out with some high-level discussion on the
software required to enable this work and this level of generality.
2.1 Mathematical Tools
The key mathematical concept underlying efficient modal solutions and effective
modal system definitions is found in the generalized eigenvalue problem, a classic
generalization of the ordinary eigenvalue problem in linear algebra.1 First, however,
we must establish why we should use the generalized eigenvalue problem in the first
place.
One of the most important characteristics of modal analysis of complex phenom-
ena is simplicity. The simplicity arises from two distinct characteristics of a useful
1Its solutions, however, should be distinguished from those of the degenerate ordinary eigenvalueproblem, in which the algebraic multiplicity or the geometric multiplicity of an eigenvalue is not1[93].
17
modal decomposition. First, the modes are orthogonal according to some meaningful
functional. Orthogonality allows the modes to be considered as separate or uncou-
pled, greatly simplifying both the mathematics and the concepts. Second, the modes
are specific to a given structure’s geometry and its material composition, but are also
excitation-independent. This quality allows the modes to satisfy the particular phys-
ical constraints of the problem so that attention is directed at understanding their
action in determining the system response given a certain system input.
2.1.1 Notation
The mathematical notation used throughout this dissertation is usually standard in
finite-dimensional linear algebra, but it is good to define the notation for clarity.
A matrix ”M” is denoted as [M ], while a column vector ”x” is denoted as x. The
transpose operation is denoted as (·)T (e.g., [M ]T or xT ), while the Hermitian (com-
plex conjugate) transpose is denoted as (·)H . Some inner products have a subscript Σ
(c.f. section 2.2.1): this implies a special inner product defined over the infinite far-
field sphere and not an inner product over a finite-dimensional vector space. Without
subscript, a ”standard” inner product is defined as⟨a, [M ]b
⟩≡ aH [M ]b. Complex
conjugation is denoted as (·)∗ and the imaginary number is j =√−1.
Unless otherwise noted, an ejωt time convention is assumed throughout, where
ω = 2πf denotes the radial frequency (units of radians) and f the linear frequency
(units of Hertz).
2.1.2 Generalized Rayleigh Quotient
To compute modes with the above characteristics for an antenna system, it is useful to
have modes which are orthogonal in two domains: the source domain, and the field
domain. In particular, this suggests that the modal sources (usually currents) are
18
orthogonal with respect to an inner product defined over the volume of the antenna,
while the modal fields (electric or magnetic or both) are orthogonal with respect to
a different inner product defined remotely from the antenna extents.
If the modal orthogonality is determined by two inner products, then the modes
may be defined as successive stationary extrema of some (generalized) Rayleigh quo-
tient:
ρ( ~J) =
⟨~J,N ~J
⟩S⟨
~J,D ~J⟩S
where⟨~J,A ~J
⟩S
is some inner product defined over the source domain S with some
associated operator A. Either N or D, or both operators are related to the field
domain. It is extremely advantageous numerically to define the modes using this
particular Rayleigh quotient because it is directly related to a generalized eigenvalue
problem under certain conditions. The physical meaning of the modes is derived from
the physical meaning of the two operators N and D, as well as their ratio.
In this work, the operators N and D are numerically approximated using finite-
dimensional matrices [N ] and [D], usually through a boundary element method. In
this case, the Quotient reduces to
ρ(J) =
⟨J , [N ]J
⟩⟨J , [D]J
⟩ (2.1.1)
where we now use the standard inner products defined in Section 2.1.1.
2.1.3 Lagrange Multipliers
The stationary points of Eq. 2.1.1, like in some variational problems, can be computed
by maximizing/minimizing the functional f(J) =⟨J , [N ]J
⟩subject to the constraint
that⟨J , [D]J
⟩= C. This problem may be stated using the method of Lagrange
multipliers:
f(J) =⟨J , [N ]J
⟩S− λn(
⟨J , [D]J
⟩S− C) (2.1.2)
19
provided that [N ] and [D] are Hermitian matrices and [D] is a positive-definite matrix.
These conditions come from two limitations to the method of Lagrange multipliers.
First, the functional f in the method of Lagrange multipliers is must be real. Second,
the constraint functional must have a minimum (i.e. the constraint must be bounded
from below).
The solutions to this Lagrange multiplier problem can be shown to be equivalent
to the generalized eigenvalue problem. The following proof of this connection is
provided assuming that [N ] and [D] are complex Hermitian matrices and J is real.
These assumptions clarify the concepts in the proof, but it can be extended to complex
J using the concept of the complex gradient operator [94], but it is an exercise left
to the reader.
Before the proof, one definition and one theorem must be established.
Definition 2.1.1. Let f(x) be a functional operating on an N dimensional real
vector space, where x ∈ RN . Then, the gradient of f(x) is given as a column vector
(in contrast to the closely associated gradient operator, which produces a row vector):
∇xf(x) =∂f
∂x=
∂f/∂x1
...
∂f/∂xN
Theorem 2.1.2. Let [A] ∈ RNxN , x ∈ RNx1, and f(x) = 〈x, [A]x〉. Then,
∂f
∂x= ([A] + [A]T )x
20
Proof. Begin by expanding f(x):
f(x) =N∑m
N∑n
xmxnAmn
=N∑m
xm
(N∑n
xnAmn
)
= x1
(N∑n
xnA1n
)+ x2
(N∑n
xnA2n
)+ . . .+ xN
(N∑n
xnANm
)Now compute the gradient with respect to x using Definition 2.1.1:
∂f
∂x=
∑Nn xnA1n∑Nn xnA2n
...∑Nn xnANn
+
x1A11 + x2A21 + . . .+ xNAN1
x1A12 + x2A22 + . . .+ xNAN2
...
x1A1N + x2A2N + . . .+ xNANN
= [A]x+ [A]T x
= ([A] + [A]T )x
Corollary 2.1.3. Let [A] ∈ CNxN and x ∈ RNx1. Furthermore, let [R], [X] ∈ RNxN
where [A] = [R] + j[X]. Also, let f(x) = 〈x, [A]x〉. Then by Theorem 2.1.2,
∂f
∂x=([R] + [R]T
)x+ j
([X] + [X]T
)x
For a Hermitian matrix [A], Corollary 2.1.3 implies that
∂f
∂x= 2[R]x (2.1.3)
since, by definition, for any Hermitian matrix [A] = [R] + j[X], [R] = [R]T ([R] is
symmetric) and [X] = −[X]T ([X] is skew-symmetric).
Now, we are finally ready to connect the Lagrange multiplier problem to the
generalized eigenvalue problem (GEP).
21
Let J ∈ RNx1 and [N ], [D] ∈ CNxN . Furthermore, let [N ] and [D] be Hermitian
matrices and [D] be a positive definite matrix. The following functional defines the
Lagrange multiplier problem, assuming that the scalar C is a real, positive constant:
f(J) =⟨J , [N ]J
⟩− λ
(⟨J , [D]J
⟩− C
)From Equation 2.1.3, the gradient of the functional is
∂f
∂J= 2[N ]J − 2λ[D]J
At the extrema of f , ∂f∂J
= 0. Let us denote the minimum or maximum points as Jn
and the associated multiplier as λn. Evaluating the functional f at an extreme point,
we obtain:
∂f
∂J= 0 = 2([N ]− λn[D])Jn
which simplifies to the generalized eigenvalue problem:
[N ]Jn = λn[D]Jn
2.1.4 Generalized Eigenvalue Problem
The points Jn at the extrema of the Lagrange multiplier problem were shown to
reduce to a generalized eigenvalue problem
[N ]Jn = λn[D]Jn (2.1.4)
under the condition that [N ] and [D] are Hermitian matrices and that [D] is further-
more a positive definite matrix and that the extrema coordinates Jn (i.e. eigenvectors)
are real. Since the method of Lagrange multipliers optimizes a real functional, the
multipliers (i.e. eigenvalues) are also real.
Also, by the complex spectral theorem [93, pg. 296], the eigenvectors are both N
and D orthogonal. That is, for m 6= n:⟨Jm, [N ]Jn
⟩>= 0 =
⟨Jm, [D]Jn
⟩22
Finally, notice then that the eigenvectors Jn (extrema in the method of La-
grange multipliers), as used here, are equal to the stationary points of the generalized
Rayleigh quotient 2.1.1 by construction. Evaluating the Quotient at these stationary
points yields the eigenvalue:
ρ(Jn) =
⟨Jn, [N ]Jn
⟩⟨Jn, [D]Jn
⟩ = λn
2.1.5 Conceptual Implications
The generalized eigenvalue problem (GEP) has been shown to be related to both a
particular Lagrange multiplier problem (under certain conditions), which in turn is
connected to the stationary points of a generalized Rayleigh quotient problem. In
other words, the GEP computes stationary points of the Rayleigh quotient, which is
itself just another functional. If we define the Quotient in a physical way that makes
its stationary points interesting for analysis and/or design, then both the eigenvectors
and eigenvalues of the GEP gain physical meaning. The theory of characteristic modes
has used this result very effectively.
2.2 Characteristic Mode Theory
The theory of characteristic modes, as formulated in [22], and related modal analysis
systems are defined by a generalized eigenvalue problem. This section will review the
versions of characteristic mode analysis and make some generalizations about what
makes a version of characteristic mode analysis useful. There are some useful details
omitted here which are noted in the various appendices.
23
2.2.1 Classical CM
The theory of characteristic modes, also known as the theory of classical characteristic
modes (CCM) to distinguish it from later modal systems, is defined by the following
Rayleigh quotient for perfectly conducting bodies:
ρCCM( ~J) =
⟨~J,X ~J
⟩S⟨
~J,R ~J⟩S
(2.2.1)
where
~Ei = Z( ~J)
which defines Z = R + jX as the EFIE (electric field integral equation) operator
that relates the tangential component of the applied electric field to the conductor
surface current. If the Z operator is discretized using subsectional basis functions and
Galerkin’s method in a Method of Moments (MoM) [21] scheme, then, the operator
Z is effectively approximated by the symmetric (not Hermitian) NxN matrix [Z] =
[R] + j[X], where [R] and [X] are real. The applied electric field tangential to the
conductor surfaces is approximated as ~Ei → Ei, while the surface current density is
approximated as ~J → J , both N element-long column vectors.
Minimizing this functional using the discrete analogs implies the following gener-
alized eigenvalue problem:
[X]Jn = λn[R]Jn (2.2.2)
Each eigenvector (known as an eigencurrent or modal current) Jn represents a
modal surface current. Each eigenvalue represents the modal stored reactive power
24
relative to the stored radiated power, or Q/ω:
Pstored =1
2
⟨Jn, [X]Jn
⟩Prad =
1
2
⟨Jn, [R]Jn
⟩λn =
Pstored
Prad
=
⟨Jn, [X]Jn
⟩⟨Jn, [R]Jn
⟩Since both [R] and [X] are real symmetric matrices by construction and [R] is postive-
definite2, both the eigenvalues and eigenvectors are real. More importantly, the eigen-
vectors simultaneously diagonalize both [R] and [X]. Since the eigencurrents are not
unique without some normalization, CCM normalizes each eigencurrent to radiate
unit power:⟨Jm, [R]Jn
⟩= δmn. This functional physically represents the real power
shared between two eigencurrents Jm and Jn on a lossless radiator, which implies
orthogonal modal fields [22] (see Appendix A for for a detailed exposition):
⟨Jm, [R]Jn
⟩=
1
η0
⟨~En, ~En
⟩Σ
= δmn
where ~Ek(θ, φ) is the modal far electric field radiated by ~Jk, η0 is the free-space wave
impedance, and Σ is the closed spherical surface at infinity. The leftmost equality is
guaranteed as a form of conservation of power. More specifically, it is a statement of
Parseval’s relation in that the power in the source region is equal to the power in the
far-field region, since the far-field transform is unitary for lossless media [37].
With the above modal orthogonalities, it can be shown [22, 23] that any arbitrary
surface current density coefficient vector J (called a ”current” for simplicity of refer-
ence, unless otherwise noted) may be expanded as a weighted sum of modal currents
(see Appendix C for more details):
J =N∑n
αnJn =N∑n
⟨Jn, E
i⟩
1 + jλnJn (2.2.3)
2Assuming the radiator has no cavity or internal modes, which do not radiate any power [22].
25
It follows from linearity that any far electric field may also be expanded as a weighted
sum of modal electric fields (see Appendix D for more details):
~E =N∑n
⟨~En, ~E
⟩Σ
=N∑n
αn ~En
Since both the eigencurrents and eigenvalues are real, classical characteristic mode
analysis is uniquely suited to performing a modal decomposition of antenna input
admittance [31]. In particular, classical characteristic modes can succinctly describe
the phenomena of series and parallel resonance [61].
2.2.2 Inagaki CM
Whereas classical characteristic mode theory essentially constructs an R-orthogonal
set of modes in coincident source and field regions (Σ naturally contains S), Ina-
gaki modes [35], [36] define a family of orthogonal eigenvectors in potentially non-
coincident source and field regions. Specifically, Inagaki modes are defined by the
following Rayleigh quotient:
ρICM =
⟨s( ~J), s( ~J)
⟩S⟨
~f( ~J), ~f( ~J⟩R
(2.2.4)
If we let ~f = G( ~J), then G is the operator which maps ~J , a source-dependent
variable, in the region S into the field ~f in the region R, where S and R are not
necessarily conincident. Assuming a reciprocal medium enveloping both S and R,
the operator G is symmetric: ~fA · G(~fB) = ~fB · G( ~fA). In this case, the Quotient
simplifies to:
ρICM( ~J) =
⟨~s( ~J), ~s( ~J)
⟩S⟨
G( ~J), G( ~J)⟩R
=
⟨~s( ~J), ~s( ~J)
⟩S⟨
~J, (G†G)( ~J)⟩S
(2.2.5)
where G† is the adjoint of the operator G.
26
Significantly, the region R could be in the far-field or the near-field of an antenna.
This technique has been used for near-field focusing [37], as well as pattern synthesis
[36].
For the purpose of applying this type of characteristic mode analysis to perfectly
conducting bodies, the functional may be further simplified by restricting the source
of the surface of the antenna using the EFIE:
~s→ ~Eitan = Z( ~J)
We can ease computation by approximating the operator Z using MoM and subsec-
tional basis functions, thereby transforming ~Eitan → Ei, ~J → J , and G→ [G]. Thus,
the Quotient is now:
ρICM(J) =
⟨[Z]J , [Z]J
⟩⟨J , [G]H [G]J
⟩ =
⟨J , [Z]H [Z]J
⟩⟨J , [G]H [G]J
⟩ (2.2.6)
We recognize that the stationary points of the quotient can be computed using the
following generalized eigenvalue problem (see Section 2.1):
([Z]H [Z]
)Jn = λn
([G]H [G]
)Jn (2.2.7)
Since the matrices [Z]H [Z] and [H] ≡ [G]H [G] are positive semidefinite Hermitian
matrices, the modal surface currents Jn are H-orthogonal and the eigenvalues λn are
real.
Thus, the Inagaki modes successively minimize the source field intensity with
respect to the field intensity in region R.
2.2.3 Generalized CM
Generalized characteristic modes [37], [38] are yet another type of modal analysis in
the characteristic mode family. They aim to be a more general form of the original
27
characteristic modes. They are related to Inagaki modes [37] and are defined by the
following Rayleigh quotient
ρGCM(J) =
⟨J , [X]J
⟩⟨J , [G]H [G]J
⟩ (2.2.8)
where we have used MoM to approximate the surface current density ~J as J and the
EFIE operator Z as [Z] = [R] + j[X] using the Galerkin method and subsectional
basis functions. By construction, [X] is a real symmetric matrix, while [G]H [G] is a
positive semidefinite Hermitian matrix.
It creates a set of modal fields that are orthogonal over a certain field region R
through [G], just like Inagaki modes, while also ensuring that the same modes are
X-orthogonal over the source region, just like classical characteristic modes. The
stationary points of Eq. 2.2.8 are computed through the usual associated generalized
eigenvalue problem because of the aforementioned properties of [X] and [G]H [G]:
[X]Jn = λn[G]H [G]Jn (2.2.9)
Thus, the modes successively minimize the modal reactive power with respect to the
field intensity in region R. If the region R is set to Σ, then it can be shown that
[G]H [G] = η0[R], where η0 =√
µ0ε0
is the background wave impedance and [R] is
the real part of [Z] [37]. Thus, generalized characteristic modes essentially reduce to
classical characteristic modes when the region R is set to the entire sphere at infinity.
A typical use in the literature of generalized characteristic modes [38] is to define
the operator G as:
W (θ, φ) ~E(θ, φ) = G( ~J)
where W (θ, φ) is some real weighting function on the far-field pattern ~E(θ, φ).
28
2.2.4 General Requirements of Modal Systems
From the previous sections, it may be observed that the modal systems have several
common features. First, the modal systems should feature orthogonal modes defined
over some source region. The modes are defined mathematically through a generalized
Rayleigh quotient, and more specifically, through two matrices. Both matrices must
be Hermitian and one matrix (the one in the denominator of the quotient) should be
positive definite.
Beyond mathematics, the quotient itself should be a physically meaningful ratio
of two distinct quantities. In the case of classical CM, the ratio is the power stored by
a mode relative to the power radiated by that mode. In the case of Inagaki CM, the
ratio is the field intensity at the source relative to the field intensity at some other
remote region.
These properties will be used in developing a new modal system optimized for
optimizing the mutual coupling in a multiple antenna system.
2.3 Outline of Approach
The problem considered herein is that there is a single antenna (herein termed the
source antenna) under consideration which can be changed. The number and location
of feed ports on this source antenna is unknown. The remaining structures and
antennas are fixed and are fully known. The antennas are (electrically) near each
other. The challenge is to identify the necessary geometric modifications to the source
antenna. A pair of modal systems are proposed for multiple antenna systems. The
first plays the role of classical CM, but for only one antenna in the system radiating
in the presence of the others. We shall refer to this modal system alternately as
Subsystem Classical CM (SCCM) or Radiation Modes. The second modal system
29
shall analyze the power coupled from the source antenna to another target antenna
(or antennas) in the multiple antenna system. We shall refer to this second modal
system alternately as Target Coupling CM (TCCM) or Coupling Modes.
2.4 UCM Software
Underlying all these computations is necessarily software. Whereas characteristic
mode software in the past has apparently focused on implementing a particular type
of characteristic mode analysis for a particular numerical electromagnetics solver, the
work undertaken here requires considerable generality in both the type of character-
istic mode analysis and the numerical solver.
2.4.1 Architecture
Like waveguide modes, characteristic mode-based analysis is physical, but it is usually
perceived to be tied to a particular piece of software or electromagnetic simulation
technique. While characteristic mode analysis utilizes various numerical routines to
enable general analysis of radiating structures, it is not fundamentally numerical in
nature. Thus, it is the software implementation of such analysis which can give the
analyst the first (incorrect) impression that characteristic modes are numerical in
nature.
To overcome this impression and explore new types of characteristic mode analysis,
there should be a modern software architecture which presents a unified interface to
analyze characteristic modes, regardless of the simulator which provides the raw data
for the analysis. It should be flexible with respect to the simulator, so that the
simulator becomes almost entirely abstracted away, hopefully leaving the impression
that the characteristic modes are physical and defined by a particular generalized
eigenvalue problem. Such architecture is difficult to preconceive in the abstract, but
30
after a few years and two attempts, Figure 2.1 has emerged as a useful software
architecture.
!"#$%&'()$*$!"#$%&'()*"+,-.-+,-+'/
0(#.$'-1/#(,-/2(-32"-+'1/4)(#/5('6/2$))-+'1/&+,/.&7-)+18/19+'6-1":-1/;-"<6'-,/2$))-+'1/
&+,/.&7-)+18/.%('1/=&)"($1/>$&+??-18/&+,/.-)4()#1/('6-)/6"<6-)/%-=-%/0@/&+&%91-1/
+,-./&012$3&4&$+1.)5($A)(=",-1/)&;/,&'&/,-)"=-,/4)(#/'6-/1"#$%&'()/4()/26&)&2'-)"1?2/#(,-/&+&%91"1/
!"#$%
6+7+,-$89)(:1;$
6+7<24(22&$8(:=&24;$
&!'(%
>6?@+,-$89)A:1;$
>6?@#&2.&/+,-$89)A-:;$
>.4.)($
+,-./&41)$
!"#$%&'()$B$!"#$%&'()*,-.-+,-+'/
0(#.$'-1/-"<-+=&%$-18/-"<-+=-2'()18/-"<-+.&7-)+18/&+,/#(,"B-1/#(,-/+$#5-)"+</$1"+</&/C0@/&+&%91"1/
#(,$%-/
!"#$%
9)(:1$!9.5-(:=$
&!'(%
9)A:1$!9.5-A(C$
9)A-:$!9.5-A-:$
&)*)+,%-./%012,+%3%
(45,6*%
!"#$<2&/':,:$#1D./($!"#$%&'()*"+,-.-+,-+'/
D&26/#(,$%-/,-B+-1/&/26&)&2'-)"1?2/#(,-/&+&%91"1/'9.-/E2%&11"2&%8/+-';()FG2%&11"2&%8/<-+-)&%":-,8/-'2HIH/D&26/
#(,$%-/"1/)-1.(+1"5%-/4()/,-B+"+</;6"26/C0@/J&9-)/K/&,&.'-)1/&)-/1$..()'-,/
Figure 2.1: UCM analysis software architecture
The above architecture has been implemented in MATLAB [92] and should be
compatible with versions R2008a and newer.3 It makes use of the object-oriented
facilities provided by MATLAB since R2008a, so as to more easily enforce the above
layer relationships and abstractions.
The architecture is broken down in several components, each of which is essen-
tially layered atop each other. At the lowest layer, the simulation data source is
a class which implements the abstract class interface (a guaranteed list of services
provided by all classes claiming to be valid simulation data sources) specified in
3MATLAB R2009b has a bug which makes it more difficult to use layered object-oriented archi-tectures, so it is recommended that the reader use a different version of MATLAB
31
UCMSimInterface.m. It is typically completely specific to the particular numerical
electromagnetic solver. At the time of this writing, there are four simulation engines
supported: ESP5 [95], FEKO 5 [96], Agilent Momentum [97], and a generic engine for
performing analysis on multiport Touchstone files (a common file format for storing
multiport network parameters).
In Layer 1, the eigenvalues, eigenvectors, eigenpatterns, and even characteristic
near-fields are computed. The majority of this work is defined through a simulation-
engine agnostic class UCMLayer1Foundation.m, but limited customization is offered
through specific Layer 1 classes. All such classes must satisfy the abstract inter-
face class UCMLayer1Interface.m. Examples of specific customizations are offering
features and optimizations which are specific to the particular simulation engine. Ad-
ditionally, certain features in the foundation class are configured here and user-defined
preferences set. A particularly important preference is selecting the generalized eigen-
value problem solution method. There are three possible solution methods and they
are all discussed in the subsequent section 2.4.2. Also important is code which com-
pensates for numerical inaccuracies in the eigenpattern and characteristic near-field
computations. Those compensations are discussed in D. Lastly, the eigenvalue track-
ing algorithm discussed in Chapter 4 is implemented in the Layer 1 foundation class.
To support a new simulation engine, a new data source class must be created,
along with a new Layer 1 class configuring the Layer 1 foundation class to work
properly with the new simulation engine.
While Layer 1 supports the computation of the various modal quantities, the
modal system itself is defined through the Analysis module layer. The Analysis
module defines both matrices involved in the generalized eigenvalue problem, along
with all the necessary machinery to compute those matrices if necessary. Since each
32
Analysis module can communicate the simulation data source, if needed, it is pos-
sible for it to be simulator-specific, so it can declare itself to be compatible with a
list of Layer 1 classes. As an analyst works on a problem, it may be necessary for
him or her to view the problem through the lens of several different types of char-
acteristic mode analysis types, so it is easy to switch between modules. There have
been been many Analysis modules created at the time of this writing, but they all
must obey UCMAnalysisInterface.m, supported by the common functionality found
in UCMAnalysisFoundation.m.
The Layer 2 class is responsible for performing all modal analysis functions. It
is both simulator-agnostic and Analysis-agnostic. Layer 2 cooperates with Layer
1 and the analysis module to collect all data necessary to compute quantities like
modal weights αn and total current or pattern from a weighted summation of modes.
Layer 2 is also responsible for plotting quantities such as eigenvalue magnitude λn,
characteristic angle (π−tan−1 λn), and eigencurrents in 3D. While all the lower layers
feature some ability to be reconfigured depending on the simulator or modal analysis
type, there is only one Layer 2 class.
Other classes exist which process the entire stack, such as a diagnostics mod-
ule designed to evaluate the accuracy of the computed modes UCM Diagnostics.m,
and various Layer 3 classes which implement proposed design techniques. A promi-
nent Layer 3 class implements the computer-aided feed design algorithm featured in
Chapter 5.
A notable omission here is that no graphical user interface has been designed or
implemented. In my opinion, such an interface can only come after some knowledge
of the typical CM analysis workflow is obtained. It is hoped that the UCM software
may find wider use so that such an interface may be developed.
33
2.4.2 Computing the Generalized Eigenvalue Problem
There are some numerical considerations which should be reviewed when computing
the GEP. Notice that [D] was constrained to not only be a Hermitian matrix, but
also a positive-definite matrix. If [D] ∈ CNxN is positive-definite, then by definition⟨J , [D]J
⟩> 0 ∀ J ∈ CNx1 except when J = 0 (the null set). If J = 0, then⟨
J , [D]J⟩
= 0.
From the definition of a positive-definite matrix and the spectral theorem [93, pg.
296], we may infer that all the eigenvalues of [D] should be positive, if only slightly
larger than 0. As mentioned in [23], the small eigenvalues of [D] may become slightly
negative due to numerical error. There are two ways to address this problem in the
literature: the method discussed in [23], and the method discussed in [51].
In [23], the problem is addressed for a real symmetric matrix [D]. This method
seeks to compute the eigenvalues and eigenvectors of an associated ordinary eigenvalue
problem. This associated problem’s matrix is generated by using the portion of
the [D] matrix only eigenvalues of [D] larger than some positive threshold. The
eigenvalues and eigenvectors of this associated ordinary eigenvalue problem are then
mapped back to λn and Jn. While the original GEP allows up to N modes, this
method essentially computes only a subset of those modes, which can sometimes
cause problems when performing eigenvalue tracking. In those cases, reducing the
number of modes tracked is the only solution.
In [51], the problem is addressed for a complex Hermitian matrix [D]. This method
depends upon [52] to compute something called the nearest positive definite pair
of matrices [N ] and [D]. That is, the matrix pair [N ] and [D] are both modified
according to a procedure to force [D] to be a positive definite matrix. The results of
the modified [N ′] and [D′] are mapped back to the original problem, pretending as
though [D] was a positive-definite matrix. Accuracy is reported to be substantially
34
improved for certain test cases in [51]. From our testing, this method should be
limited to minor adjustments of [D]. That is, if only a few eigenvalues of [D] were
slightly negative, then this approach should work well. If there are many eigenvalues
of [D] that are slightly negative or a few eigenvalues of [D] are quite negative, then
Harrington’s method works better, despite restricting the modal analysis to a subset
of the total number of modes.
2.4.3 Computing Modal Weighting Coefficients
Computing the modal weighting coefficients αn in the classical characteristic mode
system relies upon the well-known formula in 2.2.3. For modal systems in general,
however, a different expression was derived in C:
αn =
⟨Jn, [M ]J
⟩⟨Jn, [M ]Jn
⟩ (2.4.1)
where [M ] = [N ] or [M ] = [D] and [N ]Jn = λn[D]Jn. Another expression for
computing the modal weighting coefficients from theoretically orthogonal modal far-
fields was derived in D, along with some important numerical considerations:
αn =
⟨~En, ~E
⟩Σ⟨
~En, ~En
⟩Σ
(2.4.2)
where ⟨~EA, ~EB
⟩Σ
=
2π∫0
π∫0
~E∗A · ~EB sin θdθdφ
Again, these expressions are defined so that they are applicable to general modal
systems.
2.4.4 Performance Considerations
While the purpose of the UCM software is to provide a general framework to perform
characteristic mode analysis on arbitrary geometries using a variety of simulation
35
engines and a variety of analysis types, its performance characteristics are important,
especially considering that it is written in M, in the native language of Matlab.
After the first prototypes of the UCM software were created, several common com-
putations were profiled (i.e. eigenvalue tracking, computing αn from the total current
in classical characteristic modes, etc.). The performance was found to be predomi-
nately determined by disk I/O. The CPU performance was improved by redesigning
a few data structures.
Disk Performance
Most of the computation time for a given problem was spent on disk I/O. Specifically,
loading MoM Z matrices from disk and saving eigenvectors and eigenpatterns to disk
consumed a significant amount of time. To address this problem, a persistent as-
sociative array class interface called SimpleDatabaseInterface.m was developed. Two
classes implement the interface, FlatFileDatabase.m, a 100% native Matlab class, and
MySQLDatabase.m, a class which utilizes a MEX program to connect to a MySQL
database (local or remote). The performance of both classes was tested and it was
found that for typical problems, there was only a slight speed advantage to using
MySQL. Therefore, the entire UCM system uses FlatFileDatabase.m by default, al-
though it can be changed by simply modifying DatabasePreferences.m.
The persistent associative array class FlatFileDatabase.m trades off disk loading
times for memory consumption. By default, it is limited to consuming approximately
100 MB of RAM or the size of a single item, whichever is larger.
Visualization Performance
Since a number of routines require visualizing geometry in Matlab instead of in the
simulation data engine’s post-processing software, it was also necessary to improve
the visualization performance in Matlab. Specifically, the most convenient functions
36
available to visualize line segments and surfaces in three dimensions have poor per-
formance for arbitrary geometries using default options. For this reason, the class
Geometria.m was created. It can visualize a fairly large number of line segments and
triangle/quadralateral patches (tested for over 1000 elements) using low-level Matlab
graphics routines at interactive frame rates, assuming OpenGL hardware acceleration.
It can also import and export several different popular graphics formats.
37
CHAPTER 3
WIDEBAND CHARACTERISTIC MODE TRACKING
3.1 Introduction
The theory of Characteristic Modes (CM) [22] is a frequency-domain modal analysis
of radiating systems. Usually, the modes are numerically computed from a matrix.
They may be computed from either network Z-parameters [27] or from the Method
of Moments (MoM) generalized impedance matrix [23]. For the purposes of this
chapter, we shall simply consider the matrix [Z] as referring to the MoM matrix, but
the results apply equally well to the network parameters. We require that [Z(ω)] be
constructed such that it is symmetric, which typically implies the Galerkin method
applied to subsectional basis functions in MoM [23].
CM theory is built around a defining generalized eigenvalue problem (GEP). Al-
though the results in this chapter apply equally well to GEPs in other types of Char-
acteristic Mode analysis [36, 38], the GEP associated with classical Characteristic
Modes [22] shall be assumed:
[X(ω)]Jn(ω) = λn(ω)[R(ω)]Jn(ω) (3.1.1)
where [Z(ω)] = [R(ω)] + j[X(ω)] is the NxN complex MoM generalized impedance
matrix at radial frequency ω, Jn(ω) is the Nx1 eigenvector associated with mode
n, and λn(ω) is the eigenvalue associated with mode n. Jn(ω) is also known as the
nth eigencurrent. Since [X(ω)] and [R(ω)] are real symmetric matrices and [R(ω)]
38
is positive definite for lossless structures [23], the eigenvalues λn(ω) are real and the
eigenvectors Jn(ω) are X and R-orthogonal. Since it is an important concept, the
reader is reminded that if 〈x1, [M ]x2〉 = 0 for x1 6= x2, then the vectors x1 and x2 are
termed ”M-orthogonal.” We shall further assume that Jn(ω) is normalized such that⟨Jn(ω), [R(ω)]Jn(ω)
⟩= 1.
Notice that these results are parameterized by the angular frequency ω. More
specifically, the eigenpair λn(ω), Jn(ω) is parameterized by ω because [X(ω)] and
[R(ω)] are parameterized by ω. Since these two matrices are provided only at discrete
frequencies, practical wideband CM analysis requires the modes at some frequency
ω2 to be associated with the modes at ω1, independent of the bandwidth separating
ω1 and ω2. This process is termed modal tracking and is the topic of this chapter. It
is believed that this is the first work to discuss robust wideband modal tracking in
the context of Characteristic Modes.
3.2 Modal Tracking
3.2.1 Problem Description
To give some context of the problem, the CM spectrum of a 1.2 meter dipole without
tracking is shown in Figure 1. The dipole was divided into 33 segments, which
translated into a 32 x 32 MoM Z-matrix at each frequency. All 32 eigenpairs were
computed every 1 MHz from 50 to 500 MHz and are ordered at each frequency
according to ascending eigenvalue magnitude.
By considering the figure, there are a few properties of a successful modal tracking
algorithm which become evident. First, it is difficult to determine how many eigen-
pairs to compute at a given frequency a priori, since the number depends on how
39
the eigenvalues evolve over the entire frequency band. Naturally, it is easier to esti-
mate this value for narrow bandwidths than wide bandwidths. The most conservative
estimate is to compute and store all available eigenpairs at each frequency: in this
case, 32. Second, for accuracy considerations, it is likely that the frequency sweep
is relatively fine rather than coarse, so that the various observable modal properties
[22] may be observed to smoothly vary over the frequency band; therefore, comput-
ing many eigenpairs is intrinsically expensive, whether it is performed using standard
algorithms (such as those used by the MATLAB [92] function eig), or more recent
iterative methods [98]. This computational expense means that the modal tracking
algorithm should not add substantial computational complexity to the overall prob-
lem. Lastly, it is observed that the dynamic range of the eigenvalue magnitudes is
quite large: for this particular problem, they span -20 dB to almost 90 dB (110 dB
range). In this chapter, all eigenvalue and eigenvector computations are computed
by the MATLAB function eig.
The problem of associating eigenpairs from two related matrices is not a new
problem. In a general way, the algorithms fall into three categories: solving a system
of differential equations [99]; tracking eigenvalues [100]; and tracking eigenvectors
[101]. Of these classes of algorithms, modal tracking based on tracking eigenvalues in
Characteristic Modes cannot work, since some antennas have degenerate (or nearly
degenerate) modes when their structures feature some symmetry [56].
Between the remaining algorithm classes, it is observed that modal tracking via
the solution of differential equations for large, dense Hermitian matrices is computa-
tionally expensive. One of the reasons that the problem is computationally expensive
is that most methods perform the tracking using numerical integration of a set of dif-
ferential equations. In the case of MoM, it can be quite expensive to compute many
new matrices at arbitrary frequencies, as would be required by any useful quadrature
40
100 200 300 400 500−20
0
20
40
60
80
100
Frequency (MHz)
EV
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(d
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Figure 3.1: Dipole eigenvalue spectrum without modal tracking
algorithm. While it is possible to use interpolation methods [102, 103] to generate
an MoM matrix at some intermediate frequency between two closely spaced frequen-
cies, we have found experimentally that tracking via differential equations tends to
be computationally expensive when applied to moderately-sized matrices (e.g., 1000
x 1000). Another reason for the higher computational complexity is that eigenvalue
tracking algorithms usually assume only a few extreme eigenvalues and eigenvectors
will be tracked, which is clearly not necessarily the case, as was observed earlier in
Figure 3.1.
This chapter introduces a relatively low-complexity algorithm for robust wideband
modal tracking via eigenvectors in the context of CM. To our knowledge, it is the
first time such an algorithm has been discussed in connection with CM analysis. It is
noted that the proposed algorithm has been successfully applied to over one hundred
wideband CM analyses, varying from small [Z] (5 x 5) to moderately large [Z] (1000
41
x 1000), to verify its operation. The physical rationale for tracking eigenvectors is
that while the CM eigenvalue magnitudes can vary rapidly versus frequency (in linear
units) because they represent the ratio of stored power to radiated power at any given
frequency, the eigenvectors represent modal currents on the surface of antennas. It is
reasonable to expect that these modal currents should vary slowly versus frequency
and we have found this to be empirically true for modes with eigenvalue magnitudes
less than 60 or 70 dB, even around modal resonances. Therefore, a modal current
Jn(ω2) should resemble Jn(ω1), provided that ω1 and ω2 are sufficiently close and that
Jn(ω1) evolves into Jn(ω2).
3.2.2 Proposed Tracking Algorithm
Let us assume that we have computed the eigenpairs λn, Jn at two adjacent frequencies
at ω1 and ω2. Specifically,
[X(ω1)] [Γ(ω1)] = [R(ω1)] [Λ(ω1)] [Γ(ω1)]
[X(ω2)] [Γ(ω2)] = [R(ω2)] [Λ(ω2)] [Γ(ω2)]
where [Γ(ωk)] is the matrix whose columns are formed by the eigenvectors Jn(ωk),
and [Λ(ωk)] is a diagonal matrix whose entries are the eigenvalues λn(ωk). Also, we
assume that the eigenpairs at ω1 are ordered properly.
The tracking algorithm should track the evolution of [Γ(ω1)] into [Γ(ω2)]. There
are two general stages to the tracking algorithm: the association stage, where eigen-
vectors at ω1 and ω2 are determined to be related; and the arbitration stage, where
ambiguous relationships between eigenvectors are resolved.
Association Stage
We begin by assuming that the eigenvectors at ω1 are sorted and the eigenvectors
at ω2 are unsorted. Obviously, this requires the first set of computed eigenvectors
42
to be initially sorted according to some sort of scheme. A simple scheme is to sort
according to ascending eigenvalue magnitude (smallest to largest).
We define a correlation matrix [C] which relates the eigenvectors ω1 to ω2:
[Γ(ω2)] = [C][Γ(ω1)] (3.2.1)
Thus, [C] is given as:
[C] = [Γ(ω2)][Γ(ω1)]−1 (3.2.2)
Since [C] is potentially complex from the above definition, we define the correlation
matrix instead as:
[C] =∣∣[Γ(ω2)][Γ(ω1)]−1
∣∣ (3.2.3)
The goal of the tracking algorithm is to reduce the correlation matrix [C] to a permu-
tation matrix [P ]. If the correlation between an eigenvector at ω1 and an eigenvector
at ω2 is suitably high, then the two vectors are regarded as the same mode. The MAC
literature [104] seems to recommend a minimum correlation of about 0.9. In our im-
plementation, as long as only one eigenvector at ω2 is predominately correlated with
an eigenvector at ω1, we do not check for a minimum correlation between eigenvectors
at different frequencies, which has been successful in most cases. If more than one
eigenvector at ω2 is predominately correlated with an eigenvector at ω1, then some
arbitration process must occur. Arbitration will be discussed in the next section.
Assuming that a unique mapping is discovered from [C], [P ] may be constructed by
thresholding [C] (e.g., according to the minimum correlation value) and the tracking
algorithm is deemed successful in tracking the evolution of [Γ(ω1)] into [Γ(ω2)].
Arbitration Stage
Arbitration is necessary in cases where the matrix [C] is unable to resolve the re-
lationships among some subset of [Γ(ω1)] to some subset of [Γ(ω2)]. The ambiguity
43
may arise because of insufficiently high correlation among some set of eigenvectors
between the two frequencies or because of multiple eigenvectors at ω2 are apparently
mapping to a single eigenvector at ω1 (or vice-versa). All such problems are numerical
and not physical in nature. They arise primarily because ω2 − ω1 is too large, but
can also occur because of limited accuracy in the eigenvalue solver. The solution to
the latter problem is not explored here. While there are several possible arbitration
schemes, we use a combination of two schemes.
The first scheme, called adaptive tracking for the purposes of this chapter, is based
on the reasoning that if there is ambiguity between some set of eigenvectors at ω1
and at ω2, it would probably be resolved by making the frequency step, (ω2 − ω1),
smaller. Thus, adaptive tracking simply subdivides each frequency interval into two
pieces and applies the first stage algorithm to track the evolution of the eigenvectors
over each subinterval. if any given subinterval has some ambiguity, that subinterval is
subdivided again. This scheme requires fast matrix interpolation to generate the [Z]
matrices between ω1 and ω2, but since we already assume that the original frequency
step is somewhat small, linear interpolation is used. Lastly, adaptive tracking is
well-suited to recursive and parallel implementations.
The second scheme is based on the concept that if the source of ambiguity is
that multiple eigenvectors at ω2 are being associated with one eigenvector at ω1, the
eigenvector at ω2 with the highest correlation to the eigenvector at ω1 is deemed
to be the same mode as the eigenvector at ω1. Then, the process is repeated for
each remaining unassociated eigenvector at ω2 and ω1, with the restriction that only
unassociated eigenvectors may be associated. As an example, let us assume that
there are three unassociated eigenvectors at ω1 and three unassociated eigenvectors
at ω2. Furthermore, let us assume that the unassociated eigenvectors at ω1 are modes
1, 5, and 7, while the unassociated eigenvectors at ω2 are sorted as 1, 2, and 3. The
44
portion of the correlation matrix describing the correlation among these eigenvectors
between the two frequencies is given below:
[C] =
0.9 0.8 0.7
0.3 0.6 0.65
0.1 0.5 0.2
(3.2.4)
Recalling that [C] maps the sorted eigenvectors at ω1 to the unsorted eigenvectors
at ω2, this scheme would map J1(ω1) to J1(ω2), since 0.9, the largest value in the first
row, is in the first column. To associate J5(ω1) to a particular eigenvector at ω2, we
note that the correlation matrix of unassociated eigenvectors is now:
[C] =
0.6 0.65
0.5 0.2
(3.2.5)
Therefore, this scheme would map J5(ω1) to J3(ω2), since 0.65 > 0.6. By process
of elimination, J7(ω1) is mapped to J2(ω2).
In our algorithm, adaptive tracking is used as the arbitration scheme until the
frequency subinterval bandwidth is less than a certain value, in which case the second
scheme is applied. Specifically, the value is 0.01 MHz, which works well for many
geometries and was empirically chosen after much testing. This choice was made
by observing that when arbitration is typically required, the correlation among the
ambiguous eigenvectors was low (usually less than 0.6). Such low overall correlation
indicates that the unassociated eigenvectors at ω1 are only loosely correlated with
the unassociated eigenvectors at ω2, which implies that the frequency step (ω2 − ω1)
is too large. This problem is best resolved by the first arbitration scheme (adaptive
tracking) and not the second.
45
Complexity
It is already assumed that the generalized eigenvalue problem is solved at each fre-
quency (and that most, if not all, eigenvalues and eigenvectors are computed), so it
will not be counted here. In terms of complexity, the association stage of the pro-
posed modal tracking algorithm is asymptotically O(N3) [105], where [Z] is NxN ,
since an inversion of an NxN matrix is involved and assuming that we are tracking
all N eigenvectors. The first arbitration scheme, adaptive tracking, is also O(N3),
since it simply repeats the association stage at progressively finer frequency steps.
The second arbitration scheme (usually the rarer of the two) requires a sorting opera-
tion for each unassociated eigenvector at ω2, which varies in complexity depending on
the sort algorithm, but is typically O(Nlog(N)) [105]. Overall, the typical problem
will see a complexity of O(N3), since the typical problem uses the first arbitration
scheme most frequently. To reduce the computational complexity, one can choose
to track fewer eigenvectors over the band (say M eigenvectors). Then, the overall
tracking algorithm complexity will be reduced to O(M3) [105], assuming that we use
a least squares approach to computing [C] (i.e., pseudoinverse), since [Γ(ω1)] will be
a rectangular matrix and cannot be inverted as in equation 3.2.3.
It is arguable that an algorithm with O(N3) complexity hardly qualifies as abso-
lutely efficient, but in practice, MATLAB implementations of this algorithm and a
differential equation-based tracker [100], when applied over many antenna geometries
of varying complexity, have demonstrated that the proposed algorithm finishes faster
with lower average memory consumption. The main reason for this result is because
the differential equation-based tracker had slow convergence in all cases. While it is
possible for improved future differential equation-based trackers to have better per-
formance than the proposed algorithm, our experience has been that the proposed
tracking algorithm has reasonable performance with minimal resource consumption
46
for problems up to 2000 x 2000 in size and tracking at most 200 eigenvectors simul-
taneously.
3.3 Examples
Several geometries were analyzed using Characteristic Modes and the proposed modal
tracking algorithm. All antennas were analyzed using FEKO 5.5 [96] and MATLAB
R2009a [92] using double precision on a Core 2 Duo 2.5 GHz machine with Windows
7 Professional 64 bit and 4 GB of RAM.
3.3.1 Linear Wire Dipole
A 1.2 meter dipole antenna was simulated from 50 to 500 MHz in 2 MHz steps. It
was divided into 33 segments, which yielded a 32 x 32 [Z] matrix at each frequency.
Tracking 32 modes across the frequency band took about 10.1 seconds. The tracked
spectrum is illustrated in Figure 3.2. Demonstrating the reduced complexity of track-
ing fewer modes, tracking just 8 modes across the same frequency range took about
9.2 seconds.
From Figure 3.2, the modal tracking allows us to see that there are four modal
resonances, or frequencies when a particular mode’s eigenvalue magnitude becomes
much less than 1, between 50 and 500 MHz. After each resonant frequency, the eigen-
value magnitude of each mode tends to converge to about 5 dB. Although not easily
noticeable in Figure 3.2, the eigenvalue magnitudes are slightly separated (i.e. the
modes are not degenerate) and their magnitudes are slowly decreasing with frequency.
The same basic trend can be roughly observed with many different geometries, such
as the spherical helix discussed next.
47
100 200 300 400 500−20
0
20
40
60
80
100
Frequency (MHz)
EV
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(d
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Figure 3.2: 1.2 meter dipole eigenvalue spectrum with 32 tracked modes
3.3.2 Spherical Helix
The spherical helix antenna presented by S. R. Best [1] was analyzed from 50 to 550
MHz in 2 MHz steps. The geometry is shown below in Figure 3.3: it was divided
into 240 segments, which translated into a 250 x 250 [Z] matrix at each frequency.
Tracking 40 modes across the frequency band took 394 seconds. The tracked spectrum
is shown in Figure 3.4. Tracking just 8 of the modes across the same frequency range
took less time: 296 seconds.
Figure 3.4 demonstrates the benefit of accurate modal tracking, as it can be clearly
observed that there are three primary modal resonances within the band, with three
other minor modal resonances (frequencies where the eigenvalue magnitude has a
dramatic trough). This spherical helix antenna was designed to operate using the
mode which resonates at about 300 MHz. It can be observed that if this mode is
excited with a single feed point along with the mode which resonates at about 380
48
Figure 3.3: Spherical helix antenna geometry
MHz, there may be a parallel resonance in the input impedance [61], reducing the
usable impedance and pattern bandwidth.
3.3.3 Rectangular Microstrip Patch
A 38.4 x 29.5 mm rectangular patch on an infinite slab of 2.87 mm thick lossless FR4
dielectric (εr = 4.4) was analyzed from 1 to 5 GHz in 5 MHz steps. The geometry is
shown below in Figure 3.5: it was divided into 50 triangles and one thin wire segment,
which resulted in a 68 x 68 [Z] matrix at each frequency. Tracking 10 modes across
the frequency band took 11.1 seconds; the computed spectrum is shown below in
Figure 3.6.
As can be seen in Figure 3.6, there are a few frequencies where the eigenvalue
magnitudes are discontinuous. These are tracking errors. In this particular case, the
[Z] matrix provided by FEKO for triangle-type geometries is not exactly symmetric
49
100 200 300 400 500−20
0
20
40
60
80
100
Frequency (MHz)
EV
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(d
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Figure 3.4: Spherical helix antenna eigenvalue spectrum with 40 tracked modes
Figure 3.5: Microstrip patch antenna geometry
50
1000 2000 3000 4000 5000−10
0
10
20
30
40
50
60
70
80
Frequency (MHz)
EV
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(d
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Figure 3.6: Patch antenna eigenvalue spectrum with 10 tracked modes
theoretically [106]. Moreover, the [R] matrix is not theoretically positive definite (or
even semi-positive definite). These two problems significantly contribute to eigen-
vector errors, which naturally causes modal tracking errors. The matrices are being
forced to be symmetric before the GEP is solved, but the [R] matrix problem is not
resolved. While it is possible to use techniques discussed in [23] and [51] to repair any
numerical errors in [R] which cause it to become an indefinite matrix, they do not
precisely apply here, since the [R] matrix is indefinite by construction. Furthermore,
as can be observed in 3.6, some of the computed eigenvalues do not vary smoothly
versus frequency. While we are then able to compute the Characteristic Modes when
we force the theoretically asymmetric [Z] from FEKO to be symmetric, the resulting
symmetric [Z] does not correspond exactly to the original problem. Therefore, it is
a numerical artifact, rather than a result of the physics of that particular mode.
51
3.4 Alternative Algorithm
3.4.1 Formulation
The tracking algorithm works well for nearly all simulated geometries with only minor
tracking errors. Since the association stage performs most of the work, it is reason-
able to consider an alternate method of computing [C] which replaces the matrix
inverse with a simpler operation. It is possible to formulate [C] using the Cholesky
decomposition of [R] and the well-known Modal Assurance Criterion (MAC) [104]
(also known as direction cosines in other literature [107]). Here, we shall use the
direction cosines definition for the MAC (Eq. 13 from [107]).
We begin with the generalized eigenvalue problem evaluated at some frequency.
Note that since [R(ωk)] is theoretically positive definite, the generalized eigenvalue
problem may be transformed into an ordinary eigenvalue problem by using the Cholesky
decomposition of [R]:
Ln ≡ [A]Jn
[B] ≡ [A]−1
[X] Jn = λn [A]H [A] Jn
[X] Jn = λn [A]H Ln
[B]H [X] [B] Ln = λnLn
Thus,⟨Lm, Ln
⟩= δmn.
Since the Ln are orthonormal, we can use them to form a permutation matrix at
a given frequency: [P ] = [L]H [L]. This property seems to be a good alternative to
the correlation matrix proposed earlier. We define an alternative correlation matrix
using this property:
[C] = ([A(ω2)][Γ(ω2)])H([A(ω1)][Γ(ω1)])
52
where [R(ωk)] = [A(ωk)]H [A(ωk)] (i.e., [A] is the Cholesky decomposition of [R]) and
[·]H is the Hermitian transpose operator. To clarify the relationship of this expression
to the MAC, let [L(ωk)] = [A(ωk)][Γ(ωk)]. Then, we can rewrite [C] as
[C] = [L(ω2)H [L(ω1)]
Again, to ensure that [C] is composed of positive real numbers, we take the absolute
value instead:
[C] =∣∣[L(ω2)H [L(ω1)]
∣∣The above equation is identical to the MAC.
3.4.2 Performance
Unfortunately, while this algorithm appears to be more efficient owing to the Cholesky
decomposition [108], it has much slower convergence for a large number of tracked
modes. For example, in the case of the spherical helix antenna, the proposed algo-
rithm performed the association stage 522 times. This translated into 522 eigenvalue
computations. In contrast, the alternate algorithm performed the association stage
622 times, which translated into 1867 eigenvalue computations. The difference in
the number of eigenvalue computations between the two algorithms is primarily due
to the number of arbitration stages encountered. In this particular example, the al-
ternate algorithm entered arbitration 3.58 times more frequently than the proposed
algorithm. As a result, it took about 58 seconds longer to track 40 eigenvectors than
the proposed algorithm.
In all tested cases, the alternate algorithm used arbitration much more frequently
than the proposed algorithm. Therefore, the alternate algorithm, while promising
theoretically, actually performs more work in practice than the proposed algorithm
for a variety of Characteristic Mode analyses on practical antenna designs.
53
3.5 Summary
This chapter discussed the challenges unique to tracking the eigenvalues and eigen-
vectors in Characteristic Mode analysis over wide frequency bands. It introduced a
new modal tracking algorithm which featured relatively low complexity and has been
tested on over one hundred CM analysis projects. The algorithm is concerned with
tracking the evolution of eigenvectors over the frequency range. To our knowledge,
this is the first time such an algorithm has been discussed in the field of Charac-
teristic Modes. Three example antennas were shown and the algorithm performance
described in each case. The proposed algorithm was shown to be faster when track-
ing fewer eigenvectors. An alternative algorithm was described which promised lower
complexity and is related to eigenpair tracking efforts in other disciplines, but prac-
tically had poorer performance than the proposed algorithm and is therefore not
recommended.
For small to medium sized matrices, the proposed algorithm, as currently imple-
mented in MATLAB, has good enough performance on a modern personal computer.
For moderately large matrices (on the order of 1000 x 1000), the implementation
would need to take advantage of the algorithms inherent parallel structure to provide
tracked eigenpairs within a reasonable amount of time.
54
CHAPTER 4
COMPUTER-AIDED ANTENNA FEED DESIGN USING
CHARACTERISTIC MODES
4.1 Introduction
Antenna design is typically composed of three distinct, but related tasks: geometry
and materials definition, feed design, and impedance matching (when necessary). By
feed design, we mean the determination of the number and location of feed ports.
Usually, the first two tasks are lumped together, as optimum feed design for classical
antennas such as spiral antennas or others is part of the overall geometrical design.
The last task is sometimes considered independent of the antenna design and is well
treated by a variety of techniques for the single-port case [109].
For conformal antennas, an increasingly popular technique for both commercial
and military applications, the feed design does not necessarily follow from the ge-
ometry definition. The geometry definition necessarily depends upon the structure
to which the antenna must be conformal, thereby restricting the antenna geometry.
Depending on the structure, the antenna geometry may be substantially different
than that of any classical antenna. In these cases, the feed design can be challenging,
as the transition from guided waves to induced antenna currents may be quite novel.
Ideally, the feed design, like the overall antenna geometry should take advantage of
the underlying structure.
55
The theory of Characteristic Modes (CM), as introduced by Garbacz [20] and
reformulated for MoM by Harrington [22], has been instrumental in our research
group’s efforts to design conformal antennas for challenging platforms [79, 77]. Like
modal methods in other disciplines [11, 14], CM defines excitation-independent modes
through an (generalized) eigenvalue problem. These modes depend solely upon the
antenna geometry and materials. In CM, the modal surface current densities are or-
thogonal, which means that the total surface current density J can be expressed as a
weighted summation of the modal surface current densities Jn. Each weighting coef-
ficient describes how well a feed design excites its associated modal current. Because
of these properties and more, CM are quite suitable for designing the geometry and
sometimes allow for intuitive feed design. The next section of this chapter reviews
CM theory in more detail.
While CM theory may be used to effectively design antennas to take advantage
of challenging platforms, the second antenna design task of feed design is not always
clear. This chapter proposes a design technique which can readily provide practical
feed designs when guided by a human operator. To the best of our knowledge, this
is the first time such a technique has been proposed in the literature.
4.2 Background
Characteristic Mode theory, as formulated by Harrington and Mautz [22], [23], uti-
lizes the frequency-varying Method of Moments (MoM) generalized impedance matrix
[Z(ω)] = [R(ω)] + j[X(ω)] (with the radial frequency ω hereafter suppressed for no-
tational convenience) to form a generalized eigenvalue problem:
[X]Jn = λn[R]Jn (4.2.1)
56
While their initial treatment restricted the possible antenna and scatterer geome-
tries to perfectly conducting, non-resonant metallic structures, later developments
extended CM analyses to antennas incorporating lossless and lossy dielectrics [28],
[29]. Provided that [Z] is an NxN symmetric matrix and [R] is positive definite,
the set of eigenvalues λn is real and indefinite, the set of eigenvectors (i.e. modal
currents) Jn is real and both X and R-orthogonal, and the set of electric fields
(i.e. modal patterns) radiated by the modal currents ~En(θ, φ) are orthogonal over
the infinite sphere. Specifically, the modal current orthogonality is defined by the
following inner product:
⟨Jm, [M ]Jn
⟩= 0 for [M ] = [R] or [M ] = [X] and m 6= n (4.2.2)
where⟨a, b⟩
= aH b and (·)H indicates the conjugate transpose (i.e. Hermitian trans-
pose) of a column vector. The modal pattern orthogonality is defined by the following
inner product:
⟨~Em(θ, φ), ~En(θ, φ)
⟩Σ
=
π∫0
2π∫0
~E∗m(θ, φ) · ~En(θ, φ) sin θdφdθ = 0 for m 6= n (4.2.3)
where (·)∗ indicates conjugation.
The eigenvalue λn represents how much power is stored relative to radiated for a
particular mode:
λn =
⟨Jn, [X]Jn
⟩⟨Jn, [R]Jn
⟩ (4.2.4)
Therefore, when λn < 0, the mode is termed capacitive, while an inductive mode
features λn > 0.
57
The total surface current density J can be expressed as a weighted summation of
modal currents:
J =N∑n
αnJn (4.2.5)
where the modal weighting coefficients (also known as modal excitation coefficients)
are defined by the following relationship:
αn =
⟨Jn, E
i⟩
(1 + jλn)⟨Jn, [R]Jn
⟩ =
⟨Jn, [R]J
⟩⟨Jn, [R]Jn
⟩ (4.2.6)
We define the impressed electric field as Ei = [Z]J . Typically, it is assumed that
the modal currents Jn are normalized such that < Jn, [R]Jn >= 1. For the purposes
of this chapter, it shall be assumed that each modal current is so normalized. Addi-
tionally, the total pattern may be similarly decomposed into a weighted summation
of modal patterns:
~E(θ, φ) =N∑n
αn ~En(θ, φ)
where
αn =
⟨~En, ~E
⟩Σ⟨
~En, ~En
⟩Σ
(4.2.7)
and ⟨~EA, ~EB
⟩Σ≡
2π∫0
π∫0
~E∗A(θ, φ) · ~EB(θ, φ) sin(θ)dθdφ
where (·)∗ indicates complex conjugation.
Notice that λn, Jn, and ~En(θ, φ) depend solely on the antenna geometry
and materials because they are uniquely defined by the generalized eigenvalue problem
4.2.1, while αn depend upon all these quantities and the excitation Ei. Specifically,
since the excitation Ei depends upon the feed design, the αn describe how much
power is coupled from the feed port(s) to each modal current.
For the remainder of this chapter, it shall be assumed that the Characteristic
58
Modes are computed for perfectly conducting, non-resonant antenna structures using
subsectional basis functions and the Galerkin formulation of MoM.
4.3 Theory
After the design of the antenna geometry is complete for a given design iteration, the
characteristic modes are fixed, since the sets λn, Jn, and ~En(θ, φ) are functions
of geometry alone. this implies that the next major challenge in the antenna design is
to determine Ei. With Ei known, αn are completely determined. Ei is a function of
three variables: number of feed ports, feed port locations, and feed port excitations.
This triplet is herein referred to as the ”feed arrangement.” If we consider Ei to be
unknown, then αn is also unknown. If αn is unknown, then J (the total surface
current density) is also unknown. This makes sense, since if J is known exactly, then
Ei is determined exactly through Ei = [Z]J .
Despite the fact that the antenna feed arrangement is unknown in this scenario,
there still are some pieces of information that are known. First, if the structure is
adequately meshed with subsectional basis functions, then any realistic feed arrange-
ment will result in an Ei consisting of only a few non-zero components; that is, Ei
is a sparse vector such that 0 <∣∣Ei∣∣0<< N . Secondly, based on the CM analysis of
the structure, a few modal coefficients αn can be regarded as known, if only approx-
imately. Alternatively, these coefficients could be considered as the desired modal
coefficients of the realized feed arrangement. For example, the designer may identify
over a particular frequency range that modes 1 and 3 should be strongly excited,
while modes 2 and 4 should be suppressed. Thus, one possible approximation for
59
some of the αn is:
α1(ω) ≈ 1 α2(ω) ≈ 10−3010
α3(ω) ≈ 1 α4(ω) ≈ 10−3010
It is important to recognize that since Ei is regarded as unknown, αn(ω) is
unknown as a whole. That is, the designer can, at best, have only approximate
knowledge of αn(ω). In summary, the two pieces of knowledge that the designer
has of the desired feed arrangement are that Ei is a sparse vector (and probably
frequency-independent), and some properties of the desired total current J(ω) are
approximately known (by means of a few approximately known αn(ω)).
4.3.1 Mathematical Formulation
To formulate the problem in mathematical terms, we begin by specifying the as-
sumptions. Specifically, we assume to approximately know K ≤ N modal weighting
coefficients at P discrete frequencies. We also assume that there are at most M suit-
able locations at which feed ports may be placed on the mesh. Note that M may be
greater than N for certain meshes, since N is the number of basis function coefficients
which are not necessarily associated with every possible feed port location.
From the definition of the modal weighting coefficient 4.2.6, we have at each
frequency:
α = [Jn]H([R]J)
where α is the Kx1 column vector of approximately known modal coefficients, and
[Jn] is the NxK matrix whose columns are formed from the eigencurrents associated
with α.
The total current J may be expressed as:
J = [Z]−1Ei = [Y ]Ei
60
Let [F ] be the NxM frequency-independent matrix, which maps the M -element port
voltage vector γ to Ei:
Ei = [F ]γ
Thus, at each frequency, we finally have the following system of equations:
α = [Jn]H [R][Y ][F ]γ
For clarity, let
[A] := [Jn]H [R][Y ][F ]
where [A] is a KxM matrix. If we assume that the designer has chosen K < M , then
α = [A]γ (4.3.1)
is an underdetermined system of equations. If we assume that the designer has only
a vague sense of how many potential feed ports are required to realize the desired α,
it is a good assumption that 0 < |γ|0 << M (i.e. that γ is sparse).
Therefore, the ideal solution to solving 4.3.1 is to compute γ at each frequency
such that it is the sparsest possible vector consistent with the observation vector α.
Unfortunately, this is a very difficult problem computationally [110].
Drawing on the literature surrounding compressed sensing [111], it has been shown
that in a wide variety of practical cases, one may approximate |c|0 ≈ |c|1. In other
words, instead of solving the problem 4.3.1 using an `0 solver, we may approximately
solve it using an `1 solver. In this chapter, we chose to use a collection of `1 nu-
merical solvers packaged as YALL1 developed by Rice University [112]. It should be
noted that while no optimization techniques have been explicitly employed at this
stage in describing the proposed computer aided feed design technique, constrained
optimization is often used within the typical `1 solver.
61
Frequency (MHz)
Po
sitio
n #
Est. V vs. Frequency: Iteration 1
100 120 140 160 180 200
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4.1: Initial port voltage magnitudes versus frequency
4.3.2 Iterations Required
Note that 4.3.1 is solved at each frequency, so the port voltages γ will vary with
frequency. If we wish to develop a feed arrangement with frequency-independent
port voltages, we need to collapse the P port voltage vectors into a single voltage
vector. We break this process down into two stages: thresholding, and averaging.
Before examining the two stages, it is important to note that at each frequency
the port voltage vector γ is normalized according to its largest magnitude voltage.
Without this normalization, the iterations can fail to converge upon a single set of
frequency-independent port voltages γ.
Stage 1: Thresholding
For a particular antenna, the voltage vectors γ are visualized in Figure 4.1. By ex-
amining Figure 4.1, it is apparent that there should be three ports over the frequency
62
5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Position #
Magnitude
Voltages (linear)
Figure 4.2: Averaged port voltages
range. To make such a determination without manual intervention requires the volt-
ages to surpass some threshold or otherwise have the computer determine horizontal
lines in an image. In our implementation, we render the above data as an image
(specifically, Figure 4.1) and input it into a Sobel edge detection algorithm [113] to
obtain a suitable threshold. Naturally, other approaches can be applied to obtain a
suitable voltage magnitude threshold.
Stage 2: Averaging
The voltages shown in Figure 4.1 are averaged over frequency; the result is shown in
Figure 4.2. Using the threshold computed from Stage 1, the Q port locations at which
there are substantial port voltages are identified. Using these new port locations, a
new [F ] matrix is formed which maps from the Q port voltages γ′ to Ei. Then,
[A]′γ′ = α (i.e. Equation 4.3.1 with new [A]′ and new γ′) is solved at each frequency.
The process repeats until the port locations have converged.
63
In summary, the proposed technique computes the number and location of the
feed ports, along with a complex, frequency-independent voltage at each port, given
some approximate modal properties of the total current over one or more frequency
bands of interest. The modal properties are approximate αn for M < N modes,
which may be obtained from some desired far field pattern or a desired total current
distribution over the frequency bands of interest.
Convergence
It is possible that this process may fail to obtain a frequency-independent set of port
voltages γ because the `1 solver fails to compute an answer. We have empirically
found this problem to occur when the desired α are impossible to realize (i.e. they
are self-contradictory as a set). Further studies on convergence will be conducted in
future work.
Considerations on Choice of α
It is important to recognize that simply choosing modes whose approximate/desired
αn are large and feeding those values into the proposed procedure can fail to produce
the overall desired antenna operation. It is important to specify both the significant
modes (i.e. large |αn|) and suppressed modes (i.e. very small |αn|). Otherwise, a
realized feed arrangement may indeed excite the specified modes, but may also fail
to suppress undesirable modes.
Finally, it is important to not overly constrain the problem 4.3.1 by specifying
too many αn. Indeed, as more αn are specified, the probability increases that the
approximate modal weights are inconsistent with any realizable feed arrangement.
64
4.4 Example: Dipole Antenna
For all examples, the proposed procedure was implemented in MATLAB R2010b [92]
and FEKO 5.5 [96] was used to compute the MoM matrices for the given geometry
over a frequency range. It should be noted that while all the examples shown here
use wires, the method is not intrinsically limited to wire antennas. Antennas meshed
using metallic triangles or other surface elements can also be used, but it is more
difficult to define a port physically among such mesh elements. For simplicity of
presentation, we therefore limit the examples to using wire structures only, but future
work will take advantage of more sophisticated structures.
In this particular example, a 1.2 meter z-aligned dipole was meshed into 32 seg-
ments and simulated from 50 MHz to 500 MHz. The eigenvalue spectrum for the first
6 modes is shown in Figure 4.3.
0 100 200 300 400 500−20
0
20
40
60
80
100
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Figure 4.3: 1.2 meter dipole characteristic mode eigenvalue spectrum
65
4.4.1 Design 1
As an initial test, we tried to recover the usual center feed position associated with
the dipole’s first mode. The first dipole mode is associated with mode 1 [72] (that
is, its eigenpattern is substantially similar to the total pattern at or below the first
series resonance of a center-fed dipole). For the desired modal properties, we let:
α1(ω) ≈ 1 α2(ω) = α3(ω) ≈ 10−3010
from 50 MHz to 150 MHz. After 3 iterations, the algorithm yielded the solution
shown in Figure 4.4.
−1
−0.5
0
0.5
1 −1
−0.5
0
0.5
1
−0.5
0
0.5
Y (m)X (m)
Z (
m)
Figure 4.4: Dipole design 1
This design realizes the modal weight magnitudes shown in Figure 4.5 and pro-
duced the radiation patterns seen in Figure 4.6.
66
100 200 300 400 500−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Coeffic
ient M
agnitude (
dB
)
Realized weights
Mode 1
Mode 2Mode 3
Mode 4
Mode 5Mode 6
Mode 7
Figure 4.5: Dipole design 1 realized modal weight magnitudes
6dB
3
0
−3
−6
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
50 MHz (Total)
200 MHz (Total)
300 MHz (Total)
400 MHz (Total)
500 MHz (Total)
Figure 4.6: Dipole design 1 directive elevation patterns
67
As is seen directly from Figure 4.6 and indicated by Figure 4.5, the radiation
pattern holds according to the desired first dipole mode pattern shape up to about
300 MHz. After that frequency, the pattern degrades as higher order modes are
excited by the single feed point.
68
4.4.2 Design 2
In this design, we explore the question: what if we wanted to excite Mode 1 over
more bandwidth? Specifically, let us use the same desired αn coefficients as in Design
1, but require that they be satisfied from 50 to 400 MHz. In this case, the feed points
were computed after 4 iterations and are shown in Figure 4.7. The ports are located
at 0.21 meters on either side of the dipole’s midpoint and both are excited with 1 V.
−1
−0.5
0
0.5
1 −1
−0.5
0
0.5
1
−0.5
0
0.5
Y (m)X (m)
Z (
m)
Figure 4.7: Dipole design 2
The realized modal weight magnitudes are shown in Figure 4.8, while the pattern
at several frequencies is shown in Figure 4.9.
From Figure 4.9, it is obvious that the pattern maintains its desired single beam
at horizon up to 500 MHz, although the beam obvious squints at higher frequencies
due to the increased aperture electrical size. Comparing Figure 4.8 with Figure 4.5, it
69
100 200 300 400 500−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Coeffic
ient M
agnitude (
dB
)
Realized weights
Mode 1
Mode 2Mode 3
Mode 4
Mode 5Mode 6
Mode 7
Figure 4.8: Dipole design 2 realized modal weight magnitudes
7dB
4.5
2
−0.5
−3
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
50 MHz (Total)
200 MHz (Total)
300 MHz (Total)
400 MHz (Total)
500 MHz (Total)
Figure 4.9: Dipole design 2 directive elevation patterns
70
is clear that Design 2 excites primarily Mode 1 up to almost 500 MHz, while Design
1 excites primarily Mode 1 up to only 320 MHz. The tradeoff is that two feed points
are required for this extra bandwidth instead of one.
4.4.3 Design 3
In this design, we explore a different question: what if we wanted to excite Mode
2 (which is associated with dipole mode 2)? Specifically, let us define the following
desired αn coefficients from 50 MHz to 400 MHz:
α2(ω) ≈ 1 α1(ω) = α3(ω) ≈ 10−3010
After 4 iterations, the feed points were computed and are shown in Figure 4.10.
−1
−0.5
0
0.5
1 −1
−0.5
0
0.5
1
−0.5
0
0.5
Y (m)X (m)
Z (
m)
Figure 4.10: Dipole design 3
The feed points are located 0.3 meters on either side of the dipole’s midpoint.
One port is excited with 1V, while the other port is excited with -1V.
71
100 200 300 400 500−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Coeffic
ient M
agnitude (
dB
)
Realized weights
Mode 1
Mode 2Mode 3
Mode 4
Mode 5Mode 6
Mode 7
Figure 4.11: Dipole design 3 realized modal weight magnitudes
4dB
1.5
−1
−3.5
−6
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
50 MHz (Total)
200 MHz (Total)
300 MHz (Total)
400 MHz (Total)
500 MHz (Total)
Figure 4.12: Dipole design 3 directive elevation patterns
72
Examining both the realized modal weights in Figure 4.11 and the pattern cuts
in Figure 4.12, it is clear that the second dipole mode has been well-excited by the
feed arrangement over the entire frequency range of 50-500 MHz.
4.5 Example: Folded Spherical Helix Antenna
In this example, a more challenging wire geometry designed by S.R. Best from [1] is
examined. The four-arm folded spherical helix antenna is shown in Figure 4.13 and
its corresponding characteristic mode eigenvalue spectrum is shown in Figure 4.14.
−0.05
0
0.05 −0.05
0
0.05
−0.05
0
0.05
Y (m)X (m)
Z (
m)
Figure 4.13: Four-arm folded spherical helix antenna geometry from [1]
The method proposed in [1] to feed this antenna is to feed at the midpoint of one
of the arms (horizontal plane of symmetry). With this single feed point, the input
resistance is approximately 50 Ohms and there is very little reactance at 300 MHz,
73
200 400 600 800−20
0
20
40
60
80
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Mode 13
Mode 14
Mode 15
Mode 16
Mode 17
Mode 18
Mode 19
Figure 4.14: Four-arm folded spherical helix antenna eigenvalue spectrum
while producing an approximately omnidirectional pattern about horizon. The input
S11 is shown in Figure 4.15, along with the impedance Q (computed using the formula
from [114]) shown in Figure 4.16. Additionally, pattern cuts at several frequencies
in Figure 4.17. Notice that the pattern shape degraded unacceptably somewhere
between 400 and 500 MHz..
4.5.1 Design 1
In this design, we would like to find the feed point(s) to implement the pattern
generated by a short dipole, shown in Figure 4.18, between 300 and 500 MHz.
From the desired pattern, equation 4.2.7 is used to compute the αn for modes 1-4,
7, and 19. These desired αn are shown in Figure 4.19. This particular list of mode
numbers were obtained by both observing the modes with large (1,7, and 19) and
very small magnitudes (2, 3, and 4) in Figure 4.19.
74
200 300 400 500 600 700 800−10
−8
−6
−4
−2
0
Frequency (MHz)
S11 (
dB
)
S11 (dB) vs. Frequency
Figure 4.15: Folded spherical helix input reflectance versus frequency
200 300 400 500 600 700 8000
200
400
600
800
1000
Frequency (MHz)
Q
Q vs. Frequency
Figure 4.16: Folded spherical helix impedance Q versus frequency
75
4dB
−1.25
−6.5
−11.75
−17
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
200 MHz (Total)
300 MHz (Total)
400 MHz (Total)
500 MHz (Total)
Figure 4.17: Folded spherical helix directive elevation patterns
2dB
−0.5
−3
−5.5
−8
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
V. Gain (dB) @ φ=0°
Figure 4.18: Desired omnidirectional pattern
76
300 350 400 450 500−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
Frequency (MHz)
Coeffic
ient M
agnitude (
dB
)
Desired weights
Mode 1
Mode 2
Mode 3Mode 4
Mode 7
Mode 19
Figure 4.19: Folded spherical helix design 1 desired modal weight magnitudes
Iteration 1 is shown in Figure 4.20, while iteration 3 is shown in Figure 4.21.
The normalized port voltage magnitudes are shown here in relief instead of a two-
dimensional representation to emphasize the large voltage magnitudes (the width of
each peak is very thin).
After 3 iterations, several ports are located and their respective port excitation
voltages computed. The locations are shown in Figure 4.22. The ports at the top of
the helix are excited with 1V, while the ports at the bottom of the helix are excited
with -1V.
This solution realizes the modal weights shown in Figure 4.23. Various pattern
cuts are shown in Figure 4.24. In contrast to the earlier example, note that the
pattern does not degrade appreciably until about 550 MHz.
This is certainly a dramatic improvement in pattern bandwidth compared to the
original design, but unfortunately, it requires several feed points, which implies a
77
Figure 4.20: Folded spherical helix port voltages for solution iteration 1
Figure 4.21: Folded spherical helix port voltages for solution iteration 3
78
−0.05
0
0.05 −0.05
0
0.05
−0.05
0
0.05
Y (m)X (m)
Z (
m)
Figure 4.22: Folded spherical design 1
200 400 600 800−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (MHz)
Co
eff
icie
nt
Ma
gn
itu
de
(d
B)
Realized weights
Mode 1Mode 2Mode 3
Mode 4Mode 5Mode 6Mode 7Mode 8
Mode 9Mode 10Mode 11Mode 12
Mode 13Mode 14Mode 15Mode 16Mode 17
Mode 18Mode 19
Figure 4.23: Folded spherical helix design 1 realized modal weight magnitudes
79
3dB
0.5
−2
−4.5
−7
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
200 MHz (Total)
400 MHz (Total)
500 MHz (Total)
600 MHz (Total)
Figure 4.24: Folded spherical helix design 1 total elevation gain patterns
potentially complex feed network. Examining the overall distribution of feed voltages,
however, suggests an improved feed design. Since the ports at the top of the helix
are excited out of phase with the bottom ports, it suggests that we could introduce
an axial wire connecting the top and bottom of the helix. Then, we could feed the
axial wire at its center.
4.5.2 Design 2
An improved design suggested by the feed arrangement of Design 1 is shown below
in Figure 4.25.
Using this feed arrangement, we obtain the S11 shown in Figure 4.26, the Q shown
in Figure 4.27, and the pattern shown in Figure 4.28. The pattern does not degrade
significantly until about 780 MHz.
Comparing the Q of the new design in Figure 4.27 with the Q of the original
80
−0.05
0
0.05 −0.05
0
0.05
−0.05
0
0.05
Y (m)X (m)
Z (
m)
Figure 4.25: Folded spherical helix design 2 (with axis)
200 300 400 500 600 700 800−30
−25
−20
−15
−10
−5
0
Frequency (MHz)
S11 (
dB
)
S11 (dB) vs. Frequency
Figure 4.26: Folded spherical helix design 2 input reflectance versus frequency
81
200 300 400 500 600 700 8000
50
100
150
200
250
Frequency (MHz)
Q
Q vs. Frequency
Figure 4.27: Folded spherical helix design 2 impedance Q versus frequency
2dB
−1
−4
−7
−10
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
T. Gain (dB) @ φ=0°
200 MHz (Total)
600 MHz (Total)
700 MHz (Total)
750 MHz (Total)
800 MHz (Total)
Figure 4.28: Folded spherical helix design 2 total elevation gain patterns
82
design in Figure 4.16 demonstrates that the impedance Q is substantially reduced
from 300 to 700 MHz. This generally implies that using one or more matching
networks (of useful bandwidth), it may be possible to operate the original design
from 300-400 MHz, while the new design can operate from 300 to 600 or 700 MHz.
The comparison of the original design with this new axial design is discussed in more
detail in separate work [78, 115].
4.6 Summary
In this chapter, a new design technique, which readily provides practical feed de-
signs with minimal guidance from a human operator for arbitrary PEC structures,
was introduced. Its theoretical basis relies upon the fact that any practical feed ar-
rangement for an antenna must be sparse relative to the number of basis functions
meshing the same antenna. This simple fact combined with some characteristic mode
information allows us to construct an underdetermined system of equations and use
an `1 solver to solve it. From this solution, we can obtain the number, locations,
and excitation voltages of one or more feed ports to realize certain modal properties
over some specified frequency range, inspired by results from the field of compressed
sensing. From many practical designs not shown here, it was verified that using an
`2 solver in place of the `1 solver does not generally converge in a realistic solution.
There are some important properties of this procedure which are fundamental to
its function. The fact that this problem comes down to an underdetermined system of
equations allows us to use an `1 solver is certainly critical to the algorithm’s efficiency,
but more interesting is that we have an underdetermined system of equations at all.
The system of equations arises because the proposed method uses multiple properties
of the desired current in the same domain (i.e. the CM domain). Using modal
properties of the desired current allows an almost qualitative description of each
83
mode’s contribution to the desired current to translate into a practical feed design.
This quality of the proposed method is extremely useful in exploring the design space
and finalizing designs.
84
CHAPTER 5
SYSTEMATIC MUTUAL COUPLING REDUCTION
THROUGH MODAL ANALYSIS
5.1 Introduction
Multiple antenna systems, such as antenna arrays, (multiple input multiple out-
put) MIMO systems, direction-finding systems, and multiple protocol systems (like a
802.11b antenna coexisting with a Bluetooth antenna) are important applications of
antenna technology in today’s marketplace. Especially important are the emerging
MIMO systems, since they promise not only improved real-world bandwidth, but also
improved signal strength in complex multipath environments [116].
Mutual coupling among these multiple antennas, whether they operate at the
same center frequency or different frequencies, is inherent to multiple antenna sys-
tems. While some of these systems, especially arrays [117], can be designed to take
advantage of mutual coupling, others, such as MIMO or direction-finding systems,
usually find mutual coupling to be detrimental. In direction-finding systems, it is
possible to process the received signals to compensate for mutual coupling, a topic
which has been extensively reported upon in the literature [85], [86]. In MIMO sys-
tems, mutual coupling primarily causes a reduction in the overall system efficiency,
which reduces the Mean Effective Gain (MEG) and therefore decreases the link ca-
pacity [118]. The drop in efficiency is attributable to the power absorbed by one or
85
more unintended antennas relative to the power absorbed by the intended antenna
(i.e. the antenna whose radiation pattern is best suited to the incident wave). Thus,
the problems caused by mutual coupling cannot all be removed using post-processing,
since some are inherently physical.
As with single antennas, it would be useful to use Characteristic Mode theory
to systematically design multiple antenna systems. Since the chief problem unique
to multiple antenna systems is mutual coupling (this is in addition to the recog-
nized problems of single antenna systems), whatever proposed methodology should
be both general and somehow take into consideration mutual coupling. Comput-
ing the Classical Characteristic Modes of these multiple antenna systems is entirely
straightforward using the many works referenced in the Introduction chapter, assum-
ing the Method of Moments (MoM) formulation produces a theoretically symmetric
generalized impedance matrix [Z], but it fails to explicitly take into consideration
the phenomenon of mutual coupling. That is, Classical Characteristic Mode (CCM)
theory describes the radiation performance of the entire multiple antenna system,
but gives no explicit information on the performance of each constituent antenna.
An approach describing the performance of each individual antenna operating in the
presence of the other antennas is necessary when operating the system as a collection
of independent receivers (as in the MIMO and direction-finding cases) rather than as
a single multiport aperture.
In this chapter, two new modal systems are proposed, along with a method to
relate them. The first, formally termed Subsystem Classical Characteristic Modes
(SCCM) and informally as Radiation Modes, describes the radiation performance of
a single antenna radiating among several other terminated antennas or structures.
The second, formally termed Target Coupling Characteristic Modes (TCCM) and in-
formally as Coupling modes, describes a generalized form of mutual coupling between
86
a source antenna and one or more target antennas or structures. By decomposing
one set of modes in terms of the other, we can understand how an efficient radiation
mode may impact mutual coupling, allowing the designer to systematically trade off
radiation performance against mutual coupling for general metallic structures.
This chapter is divided into three major sections. First, the theory and properties
of each modal system are stated. Next, a few multiple antenna systems are analyzed
using the pair of modal systems. Lastly, some important conclusions a particular
multiple antenna system described in the literature will be obtained using this new
theory, leading to three new designs which minimize the mutual coupling among the
member antennas.
5.2 Theory
In systems employing multiple antennas, there are several unique considerations that
can otherwise be ignored in single antenna systems: antenna system excitation and
mutual coupling.
In a multiple antenna system, all ports on all antennas may be connected to the
same transmitting generator or they may all be connected to the same receiver. In
this case, when one port is being driven, the remaining ports are also being driven,
so using active (or driving point) input impedance to characterize the impedance at
each port is necessary. When the antennas are configured in this manner, they are
typically being used as an array. When each antenna or port is connected to its
own receiver/generator, then when that antenna or port is driven, the other anten-
nas/ports will appear to be terminated with loads. When the antennas are configured
in this manner, they are typically being used either independently or collectively for
direction finding or MIMO applications.
Mutual coupling is a physical property of both multiple antenna systems and
87
single antenna systems radiating in the presence of other (parasitic) structures. It
fundamentally describes the power transferred from one antenna to one or more other
antennas. Physically, one can think of mutual coupling as inducing some current from
some source antenna onto another antenna (especially near its ports).
In this chapter, we consider two proposed modal systems: Radiation Modes and
Coupling Modes. The radiation modes identify the modal currents on the source
structure which radiate efficiently, and the coupling modes identify the modal currents
on the source structure which have minimal coupling to the target structure.
To clarify this discussion, a preliminary discussion is necessary. Partitioned ma-
trices are useful to simplify the mathematical and conceptual description of a system
of objects. In this section, we are examining the interactions among two structures.
In this dissertation, the concept of a structure is not limited simply to a distinct
connected set of elements, but can also describe a group of disconnected elements as
well. For examine, the term ”structure” could be applied to describe an entire dipole,
as well as an Yagi-Uda antenna, which is composed of several disconnected elements.
On the other hand, ”structure” may also be used to describe only a portion of a
dipole.
5.2.1 Partitioned Matrices: Two Structures
In order to discuss the notion of mutual coupling between a source antenna and other
antennas or structures (its environment) using the mathematical framework of MoM,
we need to utilize the concept of partitioned matrices.
Without loss of generality, the MoM generalized impedance matrix [Z] may be
partitioned into 4 block matrices:
[Z] =
[Z11] [Z12]
[Z21] [Z22]
88
where [Z11] is the PxP generalized impedance matrix of the source antenna alone
in free space without neighboring structures, [Z22 is the QxQ generalized impedance
matrix of the other structures alone in free space without the source antenna, and [Z12
(PxQ) and [Z21] (QxP ) describe the generalized coupling between the source antenna
and its environment. Because of reciprocity and the symmetry of [Z], [Z21] = [Z12]T .
More generally, we can also partition the excitation coefficients vector Ei and the
total surface current coefficient vector J :
Ei = [Z]J
Ei(1)
Ei(2)
=
[Z11] [Z12]
[Z21] [Z22]
J (1)
J (2)
If we require that the total impressed tangential electric field on the environmen-
tal structures is zero (i.e. Ei(2) = 0), we will compute the effective [Z] matrix for the
source antenna in which it is operating in the presence of the environmental struc-
tures rather than free space (i.e. [Z11]. In other words, we will ”short” the elements
of the environmental structures together, including terminating any ports in the en-
vironment with shorts. Thus, we can derive the generalized impedance matrix of the
source antenna radiating in the presence of the environmental structures:Ei(1)
0
=
[Z11] [Z12]
[Z21] [Z22]
J (1)
J (2)
which implies that
Ei(1) = [Z11]J (1) + [Z12]J (2) (5.2.1)
0 = [Z21]J (1) + [Z22]J (2) (5.2.2)
Solving for J (2) using the second equation 5.2.2, we obtain
J (2) = −[Z22]−1[Z21]J (1)
89
Substituting for J (2) in the first equation 5.2.1, we obtain
Ei(1) =([Z11]− [Z12][Z22]−1[Z21]
)J (1)
=([Z11]− [ZM
(1)])J (1)
≡ [Z(1)]J (1) (5.2.3)
where [Z(1)] and [ZM(1)] are obvious from inspection. Notice that [Z(1)] is composed of
two components: [Z11], the source antenna in isolation, and [ZM(1)], which describes
the effect of the environment on the source antenna.
Notice that if we wish to include the effect of termination loads in the environment,
we need to simply modify [Z22] by some (probably diagonal) load matrix [ZL2] which
models the environmental terminations:
[ZML(1)] ≡ [Z12] ([Z22] + [ZL2])−1 [Z21] (5.2.4)
For the purposes of this dissertation, we shall assume that the environment is termi-
nated with arbitrary loads rather than shorts, so we shall use the following definition
of the embedded generalized impedance matrix of the source antenna [Z(1)]:
[Z(1)] = [Z11]− [Z12] ([Z22] + [ZL2])−1 [Z21] (5.2.5)
5.2.2 Partitioned Matrices: Three Structures
Here, we consider a useful generalization of the partitioned matrix theory described
in the previous section: we wish to describe the interaction between two structures
in the presence of a third. We begin with a partitioned matrix with 6 submatrices:
Ei = [Z]J
Ei(1)
Ei(2)
Ei(3)
=
[Z11] [Z12] [Z13]
[Z21] [Z22] [Z23]
[Z31] [Z32] [Z33]
J (1)
J (2)
J (3)
90
Borrowing the terms isolated and embedded from antenna array pattern analysis, let
[Z11] denote the isolated source structure generalized impedance matrix, [Z22] denote
the isolated target structure matrix, and [Z33] denote the isolated environmental
structure(s) matrix. We wish to compute the embedded two structure generalized
impedance matrix (as opposed to the embedded single structure generalized matrix
from the previous section); that is, we wish to characterize the interactions between
the source structure and the target structure in the presence of the parasitic environ-
mental structure(s). Similar to the approach taken in the previous section, we shall
require that the total impressed tangential electric field on the parasitic environmen-
tal structure is zero: Ei3 = 0. Before shorting all elements together all elements on
the parasitic environmental structure(s), we shall define [Z33L] = [Z33 + ZL3], where
[ZL3 describes the terminations (if any) on the environmental structures. Thus:
0 = [Z31]J (1) + [Z32]J (2) + [Z33L]J (3)
J (3) = −[Z33L]−1[Z31]J (1) − [Z33L]−1[Z32]J (2) (5.2.6)
Substituting the expression for J (3) into the remaining expressions for Ei(1) and Ei(2),
we have:
Ei(1) = [Z11]J (1) + [Z12]J (2) − [Z13]([Z33L]−1[Z31]J (1) + [Z33L]−1[Z32]J (2)
)=([Z11]− [Z13][Z33L]−1[Z31]
)J (1) +
([Z12]− [Z13][Z33L]−1[Z32]
)J (2)
= [Z ′11]J (1) + [Z ′12]J (2) (5.2.7)
and
Ei(2) =([Z21]− [Z23][Z33L]−1[Z31]
)J (1) +
([Z22]− [Z23][Z33L]−1[Z32]
)J (2)
= [Z ′21]J (1) + [Z ′22]J (2) (5.2.8)
91
Thus, we arrive at the desired embedded two structure generalized impedance matrix:Ei(1)
0
=
[Z ′11] [Z ′12]
[Z ′21] [Z ′22]
J (1)
J (2)
(5.2.9)
where
[Z ′11] = [Z11]− [Z13][Z33L]−1[Z31] (5.2.10)
[Z ′12] = [Z12]− [Z13][Z33L]−1[Z32] (5.2.11)
[Z ′21] = [Z21]− [Z23][Z33L]−1[Z31] (5.2.12)
[Z ′22] = [Z22]− [Z23][Z33L]−1[Z32] (5.2.13)
(5.2.14)
5.2.3 Proposed Radiation Mode System
Intuitively, we would expect that the classical characteristic modes of the source
antenna in the presence of its environmental structures are different from than the
classical characteristic modes of the source antenna in isolation. The embedded sin-
gle structure generalized impedance matrix formulation in 5.2.5 allows us a way to
compute the characteristic modes of the source antenna in the presence of the envi-
ronmental structures by defining a new modal system closely linked to classical char-
acteristic modes. We begin with the definition of the desired generalized Rayleigh
quotient:
ρSCCM( ~J) =
⟨J (1), [X(1)]J (1)
⟩⟨J (1), [R(1) ]J (1)
⟩ (5.2.15)
Since [R(1)] and [X(1)] are symmetric matrices and [R(1)] is positive definite, the
stationary solutions of this quotient are given by the following generalized eigenvalue
problem:
[X(1)]Jn(1) = λn
(1)[R(1)]Jn(1) (5.2.16)
92
where [Z(1)] = [R(1)] + j[X(1)]. Since [R(1)] and [X(1)] are real and symmetric and
[R(1)] should be positive definite, the eigenvalues λn(1) and eigenvectors Jn
(1) are real.
Additionally, the eigenvectors Jn(1) are R(1) and X(1) orthogonal. If only the source
structure is excited (i.e. Ei(2) = 0) and all environmental terminations are lossless,
then⟨Jn
(1), [R(1)]Jn(1)⟩
= 2Prad; therefore, the modal patterns ~En(1) radiated by the
modal currents Jn(1) are orthogonal under these conditions (see Appendix A for more
information).
This modal system is formally referred to as Subsystem Classical Characteristic
Modes (SCCM), or more simply, Radiation Modes.
5.2.4 Proposed Coupling Mode System
Since mutual coupling is between two antennas is defined only when the antennas have
ports and our problem definition assumes that we do not know the port location(s)
on the source antenna, mutual coupling is not strictly defined in our case. Still, we
can define a modal system which utilizes the spirit of mutual coupling. Specifically,
we wish to define modes which are orthogonal over the source structure and which
successively minimize the magnitude of the induced current on some target structure
relative to the current magnitude on the source antenna in the presence of other
environmental structures. Both the target and environmental structures are assumed
to have their ports terminated with their appropriate loads. Practically, the target
”structure” should be the areas immediately surrounding each port on the target
antenna(s).
Mathematically, this modal system will require the use of 5.2.9 under the ad-
ditional constraint that the total impressed tangential electric field on the target
structure is zero: Ei(2) = 0.
93
The target and environmental structures are assumed to have all ports terminated
in their appropriate loads, denoted by [Z22L] and [Z33L], respectively:
[Z22L] = [Z22 + ZL2] (5.2.17)
[Z33L] = [Z33 + ZL3] (5.2.18)
The current J1 may be computed from 5.2.5 and the remaining current distributions
on the target and environmental structures may be obtained from the following matrix
problem: [Z22] [Z23]
[Z32] [Z33]
J (2)
J (3)
=
−[Z21]J (1)
−[Z31]J (1)
(5.2.19)
From this mathematical setup, we may define a generalized Rayleigh quotient to
compute the ratio of the induced current distribution magnitude on a target structure
relative to the current distribution magnitude on the source structure:
ρTCCM( ~J) =
⟨J (2), J (2)
⟩⟨J (1), J (1)
⟩ =
⟨J (1), [G]J (1)
⟩⟨J (1), [I]J (1)
⟩ (5.2.20)
where
[G] = [G21]H [G21] (5.2.21)
[G21] = −[Z ′22]−1[Z ′21] (5.2.22)
[I] is the identity matrix (5.2.23)
and J (2) is the current induced on the target antenna/structure(s) by the current on
the source antenna J (1). [G21] is the matrix which maps the current from the source
antenna to the target structure. [G] is the matrix which represents the induced
current magnitude on the target structure. From the quotient, we define the coupling
modes through the following (ordinary) eigenvalue problem:
[G]Jn(1) = λn
(1)[I]Jn(1) (5.2.24)
94
Since both [G] and [I] are Hermitian matrices and [I] is always positive definite,
the eigencurrents Jn(1) are both G and I-orthogonal. Also, since the denominator
of Equation 5.2.20 just uses the identity matrix, the eigenpatterns radiated by Jn(1)
are not orthogonal. The physical meaning of the eigenvalue λn is that it is a ratio
of the induced current distribution magnitude on the target antenna/structure to
the current distribution magnitude of the associated eigencurrent Jn(1) on the source
antenna in the presence of the target and environmental structures.
5.2.5 Expressing Modes in Different Modal System
Since we have two different modal systems, it is useful to analyze how one modal
system maps to another. Here, we decompose an eigencurrent from one modal system
(say radiation modes) in terms of the modal currents in another modal system (say
coupling modes). Mathematically,
Jk,SCCM =N∑n
αknJn,TCCM (5.2.25)
where
Jk,SCCM =
Jk,SCCM (1)
Jk,SCCM(2)
(5.2.26)
Jn,TCCM =
Jn,TCCM
(1)
Jn,TCCM(2)
Jn,TCCM(3)
(5.2.27)
αkn =
⟨Jn,TCCM
(1), J∗k,SCCM(1)⟩⟨
Jn,TCCM (1), Jn,TCCM (1)⟩ (5.2.28)
and J∗k,SCCM(1) is the first partition of the column vector Jk,SCCM , assuming the
partitioning specified in 5.2.27 and not 5.2.26. It is crucial to use the right partitioning
so that J∗k,SCCM(1) and Jn,TCCM
(1) are the same length so that the inner product
95
in Eq. 5.2.28 is mathematically defined. Generally, it is recommended that the
source structure in both radiation and coupling modes is exactly the same so that
J∗k,SCCM(1) = Jk,SCCM
(1).
From Eq. 5.2.28, one can build up a ”projection” matrix:
[PSCCM−TCCM ] =
α1
1 α21 . . .
α12 α2
2 . . .
...
(5.2.29)
The entries in the kth column are the coefficients in Eq. 5.2.28 corresponding to the
decomposition of the kth radiation mode current in terms of coupling mode eigencur-
rents. A similar matrix may be computed to map from coupling modes to radiation
modes. Notice that [PSCCM−TCCM ] 6= [PTCCM−SCCM ]H .
5.3 Analysis
Before analyzing some example multiple antenna systems with the radiation and
coupling modes, it is instructive to point out what is desirable behavior in terms
of eigenvalue spectrum, as well as the projection matrix relating the two antennas.
Furthermore, we shall always define the source antenna as the same structure for
both radiation and coupling modes.
In Figure 5.1, the radiation mode eigenvalue magnitude spectrum is plotted for a
particular antenna in a multi-antenna system. The mode whose eigenvalue is denoted
by the solid blue line is a good radiator at about 2.45 GHz because it has a small
magnitude, implying that it stores relatively little reactive power relative to radiated
power. Conversely, the mode whose eigenvalue is denoted by the dashed red line is
a poor radiator over the band because it has a relatively large magnitude, implying
that it stores a lot of reactive power relative to radiated power at all frequencies over
the band.
96
2000 2200 2400 2600 2800 3000−30
−20
−10
0
10
20
30
40
50
60
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Figure 5.1: Example of desirable and undesirable radiation modes according to sub-system classical characteristic modes
2000 2200 2400 2600 2800 3000−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Figure 5.2: Example of desirable and undesirable coupling modes according to targetcoupling characteristic modes
97
Figure 5.3: Example of the projection of the coupling modes onto the radiation modesat 2.4 GHz
In Figure 5.2, the coupling mode eigenvalue magnitude spectrum for the same
source antenna in a multi-antenna system is plotted. The mode whose eigenvalue is
denoted by the solid blue line couples poorly to the target structure over the band
because it has a relatively low average eigenvalue magnitude over the band, which
implies that it induces little current on the target structure relative to the current on
the source antenna. In contrast, the mode whose eigenvalue is denoted by the dashed
red line inherently couples well to the target structure over the band because it has
a large average eigenvalue magnitude, especially at 2.40 GHz.
When we use Eq. 5.2.29 to express the first 12 radiation modes in terms of the
first 13 coupling modes at 2.4 GHz, we obtain the normalized projection matrix whose
entries’ magnitude in decibels is visualized in Figure 5.3. Examining the last column,
we see that the highest coupling mode, Mode 13, is primarily composed of radiation
98
0.5 mm
57.8 mm
20 mm
Source
Antenna
Target
Structure
Feed
Point
Y
Z
Figure 5.4: Geometry for the Dipole Pair analysis example
mode 1. Coupling Modes 9 and 11, both modes with a much lower coupling eigenvalue
magnitude, are primarily composed of radiation modes 1 and 3.
5.3.1 Example: Dipole Pair
The geometry described in this section is taken from [90]. The basic setup is two 2.45
GHz dipoles spaced 20 mm apart. Each dipole is modeled as a strip 0.5 mm wide
and 57.8 mm long. The dipoles are z-aligned and are in the y-z plane. The geometry
is shown in Figure 5.4. Each dipole is center-fed. The source antenna, also referred
to as Dipole 1, is defined as the metal contained in the blue outline (left dipole) and
the target structure is defined as the area immediately around the port (marked as a
red dot) on the second dipole, referred to as Dipole 2, outlined in red. Furthermore,
the target structure is terminated with 50Ω.
The S11 and S12 curves are shown below in Figure 5.5. Notice that the minimum
99
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S12
vs. Frequency
S11 (dB)
S12 (dB)
Figure 5.5: S11 and S12 of the dipole pair
S11 occurs at 2.45 GHz with a level of -25 dB, while the maximum S12 occurs at 2.44
GHz with a level of -5.97 dB.
Source Antenna CCM
The classical characteristic modes of just the source antenna (i.e. alone in free space)
were computed and the resulting CCM eigenvalue spectrum is shown in Figure 5.6.
The associated modal weighting coefficients are shown in Figure 5.7. As expected,
Mode 1 is resonant at 2.46 GHz and is dominant over the entire band of 2-3 GHz.
System CCM
The classical characteristic modes for the entire system were also computed and the
resulting CCM eigenvalue spectrum is shown in Figure 5.8. Notice that CCM 1
(lowest blue line) is resonant at approximately 2.35 GHz, while the CCM 2 (green
100
2000 2200 2400 2600 2800 3000−40
−20
0
20
40
60
80
100
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Figure 5.6: Classical characteristic mode eigenvalue spectrum of a single dipole
2000 2200 2400 2600 2800 3000−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 1
Mode 2Mode 3
Mode 4
Mode 5Mode 6
Mode 7
Mode 8Mode 9
Mode 10
Figure 5.7: Classical characteristic mode modal weighting coefficients of a singledipole
101
2000 2200 2400 2600 2800 3000−40
−20
0
20
40
60
80
100
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Mode 13
Mode 14
Mode 15
Mode 16
Figure 5.8: Classical characteristic mode eigenvalue spectrum of system composed oftwo dipoles
line) is resonant at 2.49 GHz, neither of which correspond to the design frequency of
2.45 GHz.
Radiation Modes
The radiation modes were computed for the source antenna and the eigenvalue spec-
trum is shown in Figure 5.9, while the associated modal weighting coefficients are
shown in Figure 5.10.
In the case of radiation modes, the resonance of Mode 1 (the dominant mode over
the entire band) is at 2.44 GHz, which is in far better agreement with the observed
minimum in the S11 level in Figure 5.5. Also, note that only odd-numbered modes
are excited, which agrees well with intuition. From the previous discussion on the
desirable behavior of this antenna, we can understand that Radiation Mode 1 is the
102
2000 2200 2400 2600 2800 3000−40
−20
0
20
40
60
80
100
120
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Figure 5.9: Radiation mode eigenvalue spectrum of Dipole 1 in the presence of Dipole2, with port 2 terminated with 50Ω
2000 2200 2400 2600 2800 3000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Curr
ent W
eig
ht M
agnitude (
dB
)
Mode 1Mode 2
Mode 3Mode 4Mode 5
Mode 6Mode 7
Mode 8Mode 9Mode 10
Mode 11Mode 12
Figure 5.10: Modal weighting coefficients of radiation modes of Dipole 1
103
2000 2200 2400 2600 2800 3000−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Mode 13
Figure 5.11: Coupling mode eigenvalue spectrum of Dipole 1 (source antenna) cou-pling to the port of Dipole 2 (target structure)
desirable mode, since it stores the least amount of power relative to the power it
radiates into its environment.
Coupling Modes
The coupling modes were computed between the source antenna and target structure.
The resulting eigenvalue spectrum is shown in Figure 5.11 and the associated modal
weighting coefficients are shown in Figure 5.12. Notice that while Coupling Mode 1 is
the most desirable, it is not excited. Instead, Coupling Modes 9 and 13 are primarily
excited, along with Mode 2 in a secondary manner. Of Modes 9 and 13, Mode 13
is the most problematic, because it has a substantially higher eigenvalue magnitude
than Mode 9, implying that it is contributing the most to the mutual coupling.
104
2000 2200 2400 2600 2800 30000
2
4
6
8
10
12
14
16
18
20
Mode 9Current Weight Magnitude (dB): 17.7472Frequency (MHz): 2450
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 13Current Weight Magnitude (dB): 19.8721Frequency (MHz): 2450
Mode 2Current Weight Magnitude (dB): 15.3488Frequency (MHz): 2450
Mode 1Mode 2Mode 3
Mode 4Mode 5
Mode 6Mode 7Mode 8
Mode 9Mode 10
Mode 11Mode 12Mode 13
Figure 5.12: Modal weighting coefficients of coupling modes
105
0.419λ00.46λ00.5λ0
X
Z
0.2λ00.2λ0
Diameter = 0.0085λ0
Source
Antenna
Target
Structure
Figure 5.13: Yagi Antenna 1
5.3.2 Example: Yagi-Uda
The two geometries described in this section are taken from the design curves in the
classic NIST document on Yagi-Uda antennas [119]. The lengths and spacing of Yagi
1 and Yagi 2 were further numerically optimized to obtain better input impedance,
so their designs diverge from those published in the NIST document. Both antennas
were designed to operate at 400 MHz (i.e. λ0 = c400 MHz
).
Yagi 1
The first Yagi-Uda antenna we shall consider is a three element Yagi and is shown
in Figure 5.13. The driver is center-fed and its port is denoted as a red dot. The
blue dashed line denotes the source antenna (the driver), while the red dashed lines
enclose the target structures (the reflector and director).
106
100 200 300 400−40
−20
0
20
40
60
80
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Figure 5.14: Yagi Antenna 1 radiation mode eigenvalue spectrum
The radiation modes were computed for this structure and the eigenvalue spec-
trum is shown in Figure 5.14. The associated modal weighting coefficients are shown
in Figure 5.15. Notice that the driver excites Mode 1 over the band and is resonant
at approximately 400 MHz, as expected.
The coupling modes were also computed for this structure and the resulting eigen-
value spectrum is shown in Figure 5.16, along with the associated modal weighting
coefficients in Figure 5.17. The coupling mode eigenvalue spectrum has a peak in
Mode 13 around 400 MHz, while the modal weighting coefficients demonstrate that
indeed Mode 13 is the dominant coupling mode at 400 MHz and it is strongly excited.
From the coupling modes, one correctly predicts that the maximum coupling between
the source antenna (driver) and target structures (reflector and director) should occur
at 400 MHz. It also clearly predicts that this strongly coupling is narrowband.
The first 13 coupling modes were projected onto the first 9 radiation modes at 400
107
100 200 300 400−100
−80
−60
−40
−20
0
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 1Mode 2
Mode 3Mode 4
Mode 5
Mode 6Mode 7
Mode 8Mode 9
Figure 5.15: Yagi Antenna 1 radiation modal weighting coefficients
100 200 300 400−150
−100
−50
0
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Mode 13
Figure 5.16: Yagi Antenna 1 coupling mode eigenvalue spectrum
108
100 200 300 400−60
−50
−40
−30
−20
−10
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 13Current Weight Magnitude (dB): −10.0536Frequency (MHz): 399
Mode 1Mode 2Mode 3
Mode 4Mode 5
Mode 6Mode 7Mode 8
Mode 9Mode 10
Mode 11Mode 12Mode 13
Figure 5.17: Yagi Antenna 1 coupling modal weighting coefficients
MHz and resulted in the normalized projection matrix whose entries are visualized
in decibels in Figure 5.18. One may note that the first radiation mode is strongly
associated with Coupling Mode 13. Coupling Modes 7, 9, and 11, which feature
weaker mutual coupling since their associated eigenvalue magnitudes at 400 MHz are
much lower than Mode 13, are all primarily composed of Modes 1 and 3 excited in
nearly equal proportion. In this case, mutual coupling is critical to establish the high
gain beam associated with a Yagi-Uda antenna.
Yagi 2
The second Yagi-Uda antenna we shall consider is a six element Yagi and is shown
in Figure 5.19. The radiation modes and coupling modes are defined in similar ways
as the last Yagi-Uda example (see Figure 5.19 for more details).
The radiation modes were again computed for this case and the eigenvalue spec-
trum is shown in Figure 5.20. The associated modal weighting coefficients are shown
109
Figure 5.18: Yagi Antenna 1 coupling mode to radiation mode projection matrix
in Figure 5.21. As expected, the first radiation mode is dominant over the band and
is approximately resonant at 396 MHz.
The coupling modes were then computed: the coupling eigenvalue spectrum is
shown in Figure 5.22, while the modal weighting coefficients are shown in Figure 5.23.
Notice that Coupling Mode 12, the mode with the highest coupling, is dominant over
the band and is especially strongly excited at 396 MHz. The coupling modes correctly
predict that the peak gain should occur at 396 MHz, and they also predict that it
will be narrowband. The radiation modes predict that the antenna should likely be
well-matched at this frequency, since Radiation Mode 1 is resonant at 396 MHz.
110
0.436λ00.452λ00.476λ0
X
Z
0.289λ00.25λ0
0.430λ0
0.406λ0
0.434λ0
0.323λ0
0.430λ0
0.422λ0
Diameter = 0.0085λ0
Target
Structure
Source
Antenna
Figure 5.19: Yagi Antenna 2
350 400 450−10
0
10
20
30
40
50
60
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Figure 5.20: Yagi Antenna 2 radiation mode eigenvalue spectrum
111
350 400 450−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 1Mode 2
Mode 3Mode 4
Mode 5
Mode 6Mode 7
Mode 8Mode 9
Figure 5.21: Yagi Antenna 2 radiation modal weighting coefficients
350 400 450−150
−100
−50
0
50
Mode 12EV Magnitude (dB): 2.95642Frequency (MHz): 400
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1EV Magnitude (dB): −146.912Frequency (MHz): 400
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Figure 5.22: Yagi Antenna 2 coupling mode eigenvalue spectrum
112
350 400 450−50
−40
−30
−20
−10
0
Frequency (MHz)
Cu
rre
nt
We
igh
t M
ag
nitu
de
(d
B)
Mode 12Current Weight Magnitude (dB): −5.38216Frequency (MHz): 396
Mode 1Mode 2
Mode 3Mode 4Mode 5
Mode 6Mode 7
Mode 8Mode 9Mode 10
Mode 11Mode 12
Figure 5.23: Yagi Antenna 2 coupling modal weighting coefficients
113
X
Z
Y
15.6 mm
100 mm
100 mm
59.6 mm
62.5 mm
10 mm
2 mm air gap between microstrip patch antenna and ground planeDipole: 59.3 mm long and 0.5 mm widePatch feed probe wire radius: 0.1 mm
Figure 5.24: Dipole beside microstrip patch geometry
5.3.3 Example: Dipole Beside Microstrip Patch
This section will examine the radiation and coupling modes of a dipole antenna in
the presence of a closely spaced patch antenna, all resonant at approximately 2.4
GHz. The geometry is shown in Figure 5.24. The dipole is the blue line beside the
microstrip patch antenna on a finite ground plane. The patch is colored green, while
the ground plane is colored grey. The microstrip patch antenna is fed using a wire
probe, which is offset from the center of the patch antenna by 10 mm, indicated in
the figure as a red dot. Both the patch and dipole were designed to resonate at 2.4
GHz when in isolation. They are spaced λ08
apart at 2.4 GHz. The feed probe is
terminated by 50Ω.
The input reflectance at port 1 (at the dipole’s center) and port 2 (the microstrip
patch antenna feed port) is shown in Figure 5.25, while the mutual coupling S12 is
shown in Figure 5.26.
114
2000 2200 2400 2600 2800 3000−30
−25
−20
−15
−10
−5
0
Dipole FeedMagnitude (dB): −29.0409Frequency (MHz): 2400
Frequency (MHz)
Ma
gn
itu
de
(d
B)
Reflectance vs. Frequency
Patch FeedMagnitude (dB): −16.5Frequency (MHz): 2310
Patch Feed
Dipole Feed
Figure 5.25: Input reflectance of dipole and patch antenna system
2000 2200 2400 2600 2800 3000−35
−30
−25
−20
−15
−10
−5
0
Frequency (MHz)
Magnitude (
dB
)
S12
vs. Frequency
S12 (dB), DipoleAndPatch_v1_NFMagnitude (dB): −2.36182Frequency (MHz): 2400
Figure 5.26: Mutual coupling between dipole and patch antenna
115
2000 2200 2400 2600 2800 3000−40
−20
0
20
40
60
80
100
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Figure 5.27: Dipole-patch radiation mode eigenvalue spectrum
Notice that although the patch feed point is physically shielded by the top plate
from the dipole and the patch is detuned somewhat by the presence of the dipole,
there is still a very significant coupling level of -2.36 dB at 2.4 GHz.
The radiation modes were computed by defining the source antenna as the dipole.
The eigenvalue spectrum is shown in Figure 5.27, while the modal weighting coeffi-
cients are shown in Figure 5.28.
From the radiation modes, it is evident that Radiation Mode 1 is dominant over
the entire band and resonates at 2.4 GHz, as expected. Notice that because the modal
resonance in Figure 5.27 is narrowband, the actual dipole resonance in Figure 5.25 is
also narrowband.
The coupling modes were also computed by defining the source antenna as the
dipole and the target structure as the microstrip patch antenna’s terminated wire
116
2000 2200 2400 2600 2800 3000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Curr
ent W
eig
ht M
agnitude (
dB
)
Mode 1Mode 2
Mode 3Mode 4
Mode 5Mode 6
Mode 7Mode 8
Mode 9Mode 10
Figure 5.28: Dipole-patch radiation modal weighting coefficients
feed probe. The resulting eigenvalue spectrum is shown in Figure 5.29, while the
associated modal weighting coefficients are shown in Figure 5.30.
Examining these two coupling mode figures, one observes that Mode 4 is the
eigencurrent which provides most of the coupling because of its substantially higher
eigenvalue magnitude. It is also one of two dominant currents, alongside Mode 2.
The peak Mode 4 excitation occurs at 2.4 GHz and is relatively narrowband, which
is in agreement with the relatively narrow peak in mutual coupling in Figure 5.26.
If one wished to reduce the coupling between the patch antenna and the dipole, one
must first analyze the coupling in terms of radiation modes; the projection of coupling
modes onto radiation modes is shown below in Figure 5.31.
From the projection, we can observe that Radiation Mode 1 is a strong component
of Coupling Mode 4. If we were to reduce the coupling by only exciting Coupling Mode
1 or 2, the matrix reports that those two modes depend significantly upon Radiation
117
2000 2200 2400 2600 2800 3000−180
−160
−140
−120
−100
−80
−60
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 4EV Magnitude (dB): −62.3784Frequency (MHz): 2400
Mode 1
Mode 2
Mode 3
Mode 4
Figure 5.29: Dipole-patch coupling mode eigenvalue spectrum
2000 2200 2400 2600 2800 30000
2
4
6
8
10
12
14
16
18
20
Mode 4Current Weight Magnitude (dB): 18.6345Frequency (MHz): 2415
Frequency (MHz)
Curr
ent W
eig
ht M
agnitude (
dB
)
Mode 2Current Weight Magnitude (dB): 17.9772Frequency (MHz): 2415
Mode 1
Mode 2
Mode 3
Mode 4
Figure 5.30: Dipole-patch coupling modal weighting coefficients
118
Figure 5.31: Dipole-patch coupling mode to radiation mode projection matrix
Mode 2, which gives a fairly different pattern than the desired Radiation Mode 1.
Therefore, if one practically tries to reduce the coupling, one must excite primarily
Radiation Mode 3 and Radiation Modes 1 and 2 weakly. Since the Radiation Mode 3
eigenvalue magnitude over the band is much larger than that of Mode 1 (see Figure
5.27), it is highly unlikely that a feed network could be designed to efficiently excite
primarily Radiation Mode 3 over Mode 1. Therefore, if the mutual coupling is to
be reduced, another method of exciting Radiation Mode 3 must be found, ideally
lowering Radiation Mode 3’s eigenvalue on the source antenna.
119
5.4 Design
In this section, three designs shall be proposed, all which reduce the mutual coupling
between a pair of dipoles (the same geometry as in the first analysis example) at 2.45
GHz using the information from radiation and coupling modes. Neither dipole will be
modified. They will be compared against a design published in the literature which
successfully reduces the mutual coupling between the same pair of dipoles.
Before embarking on the reduced mutual coupling designs, let us summarize the
critical observations made of the two dipole system after analysis using radiation and
coupling modes. Firstly, the peak mutual coupling level without any treatment (just
two closely spaced parallel dipoles) is -5.97 dB at 2.44 GHz. From the coupling modal
weighting coefficients in Figure 5.12, we observed that Modes 9 and 13 are primarily
excited at the design frequency of 2.45 GHz.
If we did not care at all about the radiation pattern, then Figure 5.32 offers us a
reasonably straightforward solution: we could attempt to excite Coupling Mode 12,
which is seen to be composed primarily of Radiation Mode 2 (similar to the second
dipole mode), and it features a substantially reduced mutual coupling level according
to the coupling eigenvalue spectrum relative to Coupling Mode 13. Of course, because
the eigenvalue magnitude associated with Radiation Mode 2 is larger than Radiation
Mode 1 around 2.45 GHz, one can expect moderate to severe mismatch at the two
required feed points. Therefore, primarily exciting Radiation Mode 2 is not practical
without geometrical modifications or loading of the source antenna.
When we examine the associated coupling eigenvalue magnitudes of Coupling
Modes 9 and 13 in Figure 5.11, we observe that Mode 13 has a substantially higher
magnitude than Mode 9, implying that the bulk of the coupling is caused by Mode
13. Thus, we have identified Mode 13 as undesirable and we wish to not excite it so
strongly.
120
Figure 5.32: Dipole 1 coupling mode to radiation mode projection matrix
To understand how to modify our excitation, let us examine the projection of the
coupling modes on the radiation modes in Figure 5.32.
From the 13th column, we observe that Coupling Mode 13 is substantially com-
prised of Radiation Mode 1, our desired radiation mode. If we now examine the 9th
column corresponding the decomposition of Coupling Mode 9 in terms of the radia-
tion modes, we see that Radiation Modes 1 and 3 must be almost equally excited.
This stands in contrast to Coupling Mode 13. So, the analysis shows us that in order
to reduce the mutual coupling between the dipoles, we must find a way to excite
Radiation Modes 1 and 3 more equally on the source antenna (Dipole 1).
Geometry Modification Through Parasitic Structures
While one could use lumped reactive loading on the source antenna [74, 75] to lower
the Radiation Mode 3 eigenvalue magnitude on the source antenna around 2.45 GHz,
it has the unfortunate side-effect of raising the Mode 1 eigenvalue magnitude at the
121
same frequency. Since the loading technique only enables the control of a single mode
at any given frequency, it is clear that loading the source antenna cannot be used to
provide equal excitation of Radiation Modes 1 and 3 on it. Simultaneously, it is also
clear that it is impossible to design a feed network to directly excite Radiation Modes 1
and 3 with equal magnitudes, since the modal weighting coefficients (see Eq. 2.2.3) are
approximately inversely proportional to the Radiation Mode eigenvalue. Therefore,
approximately equal excitation of the two modes would require an impressed Ei that
is impractical.
I propose that the addition of a neighboring parasitic structure, which when il-
luminated with a Radiation Mode 1 near-field, reflects a Radiation Mode 3 field,
thereby indirectly enhancing the excitation of Mode 3 on the source antenna. To
ensure that the Radiation Mode 3 eigenvalue on the source antenna is lowered, the
parasitic structure must be close to resonance at 2.45 GHz such that its near-field
sufficiently interacts with the source antenna (i.e. that there is some coupling between
the parasitic structure and the source antenna).
The next section analyzes a proposed design for the parasitic structure (although
its design was derived from very different principles), and the subsequent sections
show three improvements on the original parasitic structure, based on the previous
modal analysis of this dipole pair.
5.4.1 Original Design
In [90], the problem of the reducing the coupling between parallel dipoles was pro-
posed and a solution obtained. A parasitic ”top-hat” loaded dipole was inserted
between the two dipoles. Both the design discussed in this section and in subsequent
sections can be generally described by the parameterized geometry in Figure 5.33.
The two dipoles (black) are 0.5 mm wide and 57.8 mm long. The red dot denotes the
122
X
Z
Y
57.78 mm
20 mm
L1 mm
L2 mm
Black lines: Dipoles
Blue lines: Hats
Green line: Coupler
Figure 5.33: Dipole pair with parasitic structure geometry
terminated port on Dipole 2, while Dipole 1 (the source antenna) lacks a dot. Both
dipoles lie in the yz-plane, while the parasitic structure, herein termed the coupler,
lies in the xz-plane, exactly midway between the two dipoles. The ”hats” on the
coupler (blue lines) are L1 mm long and W2 mm wide. The coupler itself (green line)
is L2 mm long and W2 mm wide.
In the original coupler design, the geometrical parameters for the coupler were:
L1 = 39 mm, W1 = 4 mm, L2 = 12 mm, and W2 = 2 mm. With these parameters,
the radiation modes were computed for Dipole 1. The eigenvalue spectrum is shown
in Figure 5.34, and the associated modal weighting coefficients are plotted in Figure
5.35. The black vertical dashed line indicates 2.45 GHz. Comparing Figure 5.35 with
Figure 5.10, one can clearly observe that Radiation Mode 3 is better excited in this
design than the design without a coupler. Furthermore, comparing the two radiation
123
2000 2200 2400 2600 2800 3000−20
0
20
40
60
80
100
120
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Figure 5.34: Dipole 1 radiation mode eigenvalue spectrum with the original couplerdesign
mode eigenvalue spectrums, one can observe that the eigenvalue for Mode 3 decreased
when the coupler was inserted.
In this design, the coupler is nearly resonant at 2.45 GHz, as can be observed by
computing the radiation mode eigenvalue spectrum of the coupler, shown in Figure
5.36. The coupler is actually resonant at 2.42 GHz. The addition of the coupler
improves the isolation between the two dipoles (Figure 5.38), at the cost of input
reflectance (Figure 5.37), although it remains acceptable at -9.06 dB at 2.45 GHz.1.
The isolation improved from 6.00 dB to 12.50 dB.
1It should be noted that FEKO computed a higher input reflectance level in the presence of thecoupler than was reported by CST and measurements in [90]. The simple nature of the FEKOfeed model is likely the cause of this discrepancy. In any case, all comparisons are made withrespect to the FEKO version of the problem.
124
2000 2200 2400 2600 2800 3000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Curr
ent W
eig
ht M
agnitude (
dB
)
Mode 1Mode 2
Mode 3Mode 4Mode 5
Mode 6Mode 7
Mode 8Mode 9Mode 10
Mode 11Mode 12
Figure 5.35: Dipole 1 radiation modal weighting coefficients with the original couplerdesign
2000 2200 2400 2600 2800 3000−30
−20
−10
0
10
20
30
40
50
60
Mode 1EV Magnitude (dB): 0.772039Frequency (MHz): 2450
Mode 1EV Magnitude (dB): −23.8495Frequency (MHz): 2420
Frequency (MHz)
EV
Ma
gn
itu
de
(d
B)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Figure 5.36: Original coupler radiation mode eigenvalue spectrum
125
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S11
vs. Frequency
No coupler
Original
Figure 5.37: Original coupler design input reflectance
2000 2200 2400 2600 2800 3000−22
−20
−18
−16
−14
−12
−10
−8
−6
−4
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S12
vs. Frequency
No coupler
Original
Figure 5.38: Original coupler design mutual coupling
126
Shadow region
(Reflection from coupler attenuates source current)
Source current looks like
a blend of Mode 1 and Mode 3
Figure 5.39: How the original coupler design induces a mixture of Mode 1 and Mode3 on Dipole 1
5.4.2 Design Theory
The results from the original coupler design are very interesting, especially since
they verify that the concept that as Radiation Mode 3 is made more easily excitable
alongside Mode 1 on the source antenna, the mutual coupling is reduced. The coupler
design excites Mode 3 on the source antenna because of reflections coming from both
the coupler and the terminated dipole, as shown in Figure 5.39.
Still, the coupler is fairly bulky, primarily because of its hat. What if we shortened
L2 to make the overall design more compact? If we shorten the hat, then we can use
reactive loading to cause the coupler to resonate. The equation for reactive loading
used in [74] must be modified slightly, however. Notice that the original coupler
design did not resonate at the operating frequency, but slightly below. Thus, the
eigenvalue of the dominant mode of the coupler is non-zero (in this case, inductive).
We shall use the basic methodology reported in [74], but with a slight modification
127
of Equation 3, which was used to compute the reactive loads, so that the desired
eigenvalue may be non-zero.
Let us assume that we have chosen N ports on the coupler and have identified
the desired real current Id(ω) that we wish to enforce on the coupler using uncoupled
reactive loads at those ports. Let the N uncoupled loads be represented by the diag-
onal matrix [XL(ω)]. If λd(ω) is the corresponding desired eigenvalue for my desired
current Id(ω), then the loaded network characteristic mode generalized eigenvalue
problem is:
[X(ω) +XL(ω)]Id(ω) = λd(ω)[R(ω)]Id(ω) (5.4.1)
[XL(ω)]Id(ω) = ([λd(ω)R(ω)−X(ω)]) Id(ω) (5.4.2)
Since [XL(ω)] is diagonal, the reactance at a port i is:
XLi(ω) =
1
Idi(ω)([λd(ω)R(ω)−X(ω)]Id(ω)
)i
(5.4.3)
For Designs A-C, we defined 5 ports placed equally along the coupler’s length.
5.4.3 Design A
For Design A, we wish to load the coupler to compensate for shorter hats. In this
design, L1 = 20 mm (same as the spacing) and all other parameters are the same as
in the original coupler design. I chose to enforce a constant current distribution over
the coupler to emulate the behavior of the original coupler: Id = [1 1 1 1 1]T . With
some experimentation, I chose λd = 109.75/10. λd controls the degree of influence that
the coupler has on the source antenna. If λd is too small, then the current near the
source antenna’s feed port is affected too much and the input reflectance becomes
poor. If λd is too large, then Radiation Mode 3 won’t be sufficiently excited on the
128
2 2.2 2.4 2.6 2.8 3
0
10
20
30
40
50
60
Frequency (GHz)
X (
Ohm
s)
Reactance vs. Frequency
Design A: Port 1
Design A: Port 2
Design A: Port 3
Figure 5.40: Design A ideal loads
source antenna, thereby having only a minimal reduction in mutual coupling. For
each design, I vary λd until the input S11 level matches that of the original design at
2.45 GHz.
With these settings, the ideal load reactances were computed using Eq. 5.4.3 and
are shown in Figure 5.40. While the reactances are obviously non-Foster in nature and
are therefore difficult to directly realize, they demonstrate the potential of the design
method, always assuming that the loads may be realized over a smaller bandwidth
using realistic circuit components and topologies.
The input reflectance and mutual coupling are shown in Figure 5.41 and Figure
5.42, respectively. The isolation at 2.45 GHz improved from 6 dB to 10.89 dB, which
is less than the improvement offered by the original coupler design.
To better understand the reason why this design did not work as well as the
original coupler design, it is instructive to examine the radiation modes of the source
129
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S11
vs. Frequency
No coupler
Design A (ideal loads)
Figure 5.41: Coupler Design A input reflectance
2000 2200 2400 2600 2800 3000−35
−30
−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S12
vs. Frequency
No coupler
Design A (ideal loads)
Figure 5.42: Coupler Design A mutual coupling
130
2000 2200 2400 2600 2800 3000−20
0
20
40
60
80
100
120
Frequency (MHz)
EV
Magnitude (
dB
)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Mode 10
Mode 11
Mode 12
Figure 5.43: Design A source dipole radiation mode eigenvalue spectrum
antenna. The eigenvalue spectrum is shown in Figure 5.43, and the associated modal
weighting coefficients are plotted in Figure 5.44.
The problem in this design is that while Radiation Mode 3’s eigenvalue was low-
ered, it occurred at a higher frequency band (around 2.5-2.65 GHz) than intended,
which is due to the smaller nature of the coupler. That is, reactive loading cannot
be used to dramatically increase the influence of an electrically small structure (like
the coupler in this case).
5.4.4 Design B
In this design, we seek to improve upon Design A by using an electrically larger
coupler, but making it nearly planar. In this case, L1 = 10 mm and L2 = 57.78 mm
(the dipole’s length).
I chose to enforce a current resembling the Mode 3 eigencurrent, since I want the
131
2000 2200 2400 2600 2800 3000−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (MHz)
Curr
ent W
eig
ht M
agnitude (
dB
)
Mode 1Mode 2
Mode 3Mode 4Mode 5
Mode 6Mode 7
Mode 8Mode 9Mode 10
Mode 11Mode 12
Figure 5.44: Design A source dipole radiation modal weighting coefficients
coupler to reflect back a Radiation Mode 3 field: Id = [-0.6 0.6 1 0.6 -0.6]T . With
some experimentation, I chose λd = −101.2/10. With these settings, the ideal load
reactances were computed and are shown in Figure 5.45.
The input reflectance and mutual coupling are shown in Figure 5.46 and Figure
5.47, respectively. The isolation at 2.45 GHz improved from 6 dB to 13.63 dB, which
is an improvement over the original coupler design.
5.4.5 Design C
In this design, we seek to improve upon Design B by employing a purely planar design.
In this case, L1 = 0, and L2 = 57.78 mm (the dipole’s length).
I chose to enforce a current resembling the Mode 3 eigencurrent: Id = [-0.7 0.5 1 0.5 -0.7]T .
Deciding upon the eigencurrent values does involve some guess-and-check, as this
particular eigencurrent gave improved isolation than the eigencurrent in Design B for
132
2 2.2 2.4 2.6 2.8 3
200
300
400
500
600
700
Frequency (GHz)
X (
Ohm
s)
Reactance vs. Frequency
Design B: Port 1
Design B: Port 2
Design B: Port 3
Figure 5.45: Design B ideal loads
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S11
vs. Frequency
No coupler
Design B (ideal loads)
Figure 5.46: Coupler Design B input reflectance
133
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S12
vs. Frequency
No coupler
Design B (ideal loads)
Figure 5.47: Coupler Design B mutual coupling
this geometry. Obviously, an optimization routine could be used to compute the best
possible eigencurrent choice to maximize isolation while minimizing reflectance at the
design frequency. With these settings, the ideal load reactances were computed and
are shown in Figure 5.48.
The input reflectance and mutual coupling are shown in Figure 5.49 and Figure
5.50, respectively. The isolation at 2.45 GHz improved from 6 dB to 13.06 dB, which
is an improvement over the original coupler design, but slightly less than Design B.
134
2 2.2 2.4 2.6 2.8 3
200
300
400
500
600
700
Frequency (GHz)
X (
Ohm
s)
Reactance vs. Frequency
Design C: Port 1
Design C: Port 2
Design C: Port 3
Figure 5.48: Design C ideal loads
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S11
vs. Frequency
No coupler
Design C (ideal loads)
Figure 5.49: Coupler Design C input reflectance
135
2000 2200 2400 2600 2800 3000−25
−20
−15
−10
−5
0
Frequency (MHz)
Ma
gn
itu
de
(d
B)
S12
vs. Frequency
No coupler
Design C (ideal loads)
Figure 5.50: Coupler Design C mutual coupling
136
2300 2350 2400 2450 2500 2550 2600−25
−20
−15
−10
−5
0
Frequency (MHz)
Magnitude (
dB
)
S11
vs. Frequency
No coupler
Original coupler
Design A
Design B
Design C
Figure 5.51: Comparison of input reflectance at Port 1 for all designs
5.4.6 Comparison of Designs
The input reflectance of the various coupler designs is compared in Figure 5.51. While
at the design frequency, all the couplers feature approximately -9 dB of input re-
flectance, the no coupler case clearly performs better. This is not surprising, since
the coupler is destructively interfering with the current at the feed of the source
dipole, thereby producing mismatch. Comparing among coupler designs, however,
Design B has the smallest variation in S11 level around 2.45 GHz, implying more
accessible bandwidth. Of course, loads must be developed to access this potential
bandwidth.
The tradeoff between S11 level and isolation is apparent when examining the effect
of each coupler design on reducing the mutual coupling level near the design frequency.
Compared to the no coupler case, every design features improved isolation. Designs B
and C offer improved isolation relative to the original coupler design in a significantly
137
2300 2350 2400 2450 2500 2550 2600−35
−30
−25
−20
−15
−10
−5
0
Frequency (MHz)
Magnitude (
dB
)
S12
vs. Frequency
No coupler
Original coupler
Design A
Design B
Design C
Figure 5.52: Comparison of mutual coupling for all designs
reduced volume. Again, Design B has the smallest frequency variation about 2.45
GHz, so it has the most potential bandwidth.
Another indicator of reduced mutual coupling is the radiation efficiency. If the
source antenna is more isolated from the port on the other dipole, then less power
should be transferred to that port. Therefore, the radiation efficiency of a single
antenna radiating in the presence of the other antenna terminated should be higher
with greater isolation. The radiation efficiency of each design is shown in Figure 5.53.
The radiation efficiencies at 2.45 GHz for each design are tabulated in Table 5.1.
The source dipole is radiating in the presence of the other dipole, which is terminated
with 50Ω.
Perhaps the most interesting result is the embedded radiation patterns at 2.45
GHz, shown in Figure 5.54. While the original design and Design A both feature
138
2300 2350 2400 2450 2500 2550 26000
10
20
30
40
50
60
70
80
90
100
Eff
icie
ncy (
%)
No coupler
Original coupler
Design A
Design B
Design C
Figure 5.53: Comparison of radiation efficiency for all designs (ideal loads)
Table 5.1: Radiation efficiency of a source dipole 2.45 GHz
Design Name Efficiency (percent)
No coupler 74.76
Original 93.58
Design A 90.70
Design B 95.05
Design C 94.35
139
6dB
2
−2
−6
−10
0ο
30ο
60ο
90ο
120ο
150ο
180ο
210ο
240ο
270ο
300ο
330ο
No coupler
Original coupler
Design A
Design B
Design C
Figure 5.54: Comparison of embedded vertical gains at 2.45 GHz at θ = 0 (azimuthalcuts) for all designs
”beams” which point away from the other dipole, Designs B and C feature fear-field
beams which point towards the the other dipole.
5.5 Summary
This chapter introduced two new modal systems. One is formally termed Subsys-
tem Classical Characteristic Modes (SCCM), also called Radiation Modes. It is an
extension of Classical Characteristic Modes, but applied to antennas or structures
embedded among other structures rather than free-space. It was shown to be quite
useful in identifying the true modal resonant frequency of a structure operating in
the presence of other structures, among other applications. The second new modal
system, formally termed Target Coupling Characteristic Modes (TCCM), also called
Coupling Modes, is a modal system which can analyze the coupling between struc-
tures. It was applied to several antennas and was shown to accurately predict the
140
qualitative behavior of Yagi-Uda antennas, antennas that depend upon high amounts
of coupling to produce high gain. Finally, a projection matrix was defined, which
relates the two modal systems. The projection matrix was used to modify a multiple
antenna system to reduce the mutual coupling among the antennas by introducing a
loaded parasitic. The loads were determined based on the information provided by
the coupling modes.
I believe that this pair of modal systems could be used to at systematically analyze
coupling among structures and possibly control the coupling through the creative
application of the modal information.
141
CHAPTER 6
CONCLUSIONS
6.1 Summary and Conclusions of the Dissertation
The work considered in this dissertation is overall concerned with the systematic
analysis and design of complex antenna problems. Although such systems contain
a wealth of information, the sheer volume is overwhelming and various specific ap-
proaches have been historically applied to extract the relevant physics. This disserta-
tion applies and develops types of characteristic mode analysis through mathematics
and software to provide a general description of the physics, making clearer some key
antenna design characteristics, such as bandwidth, feed design, and mutual coupling.
6.1.1 Software
In Chapter 2, the mathematical foundation and key properties of successful modal
systems were reviewed. Furthermore, the chapter provided an overview of the software
developed in connection with this dissertation. In total, over 60,000 lines of Matlab
code were written to aid Prof. Roberto Rojas’ research group and are maintained
through a custom decentralized software update system. The Unified Characteristic
Mode (UCM) system, in particular, has been the foundational system responsible
for all characteristic mode analysis, from the computation of the modes using data
extracted from various commercial electromagnetic simulation software packages, to
142
their efficient storage (individual UCM projects can grow to be gigabytes in size), to
a plugin system concisely expressing new modal systems, all using an object-oriented
approach and in nearly all native Matlab code.
6.1.2 Wideband Mode Tracker
In Chapter 3, a wideband, robust mode tracking system was developed to automati-
cally associate modes at different frequencies in a practically computationally efficient
way, the first of its kind reported in the field of characteristic modes. Such a tracker
is essential to considering how the characteristic modes (whether they are classical,
generalized, Inagaki, or others) evolve over frequency for a given antenna or scatterer.
Especially crucial is its automated nature, since in practice, perhaps upwards of 40 or
50 modes must be tracked over hundreds or thousands of frequency points. Compar-
isons to existing trackers demonstrated that they were either too inaccurate or too
computationally expensive for this problem scale. An alternative tracking algorithm
was provided as a point of comparison for the computational cost. While not explic-
itly discussed in Chapter 3, the algorithm is foundational to Chapters 4 and 5, as the
developments in both chapters critically depend upon considering the evolution of
modes over some frequency band to derive their unique results. The tracker has been
under development since 2008 and has been applied to over 100 antennas, enabling
the author and his colleagues, Khaled Obeidat and Brandan Strojny, to understand
the frequency behavior of the characteristic modes on actual antenna designs. While
certain aspects can still be improved, its robustness has been an important contribu-
tion to both this work and that of others.
143
6.1.3 Antenna Feed Design
In Chapter 4, a semi-automated design technique was developed which computes the
number, location, and voltages of ports on a given antenna according to realize some
desired modal and bandwidth specifications. It uses the desired modal weighting
coefficients computed according to some type of characteristic mode analysis to form
an underdetermined system of equations over frequency. Using the fact that any
realistic port arrangement is sparse relative to the surface area of the antenna and
the properties of `1 solvers, discussed extensively in the emerging field of Compressed
Sensing, the procedure computes the most likely port locations to realize the desired
modal weighting coefficients over some frequency range. In fact, the frequency range
need not be contiguous, enabling the procedure to potentially compute the optimum
feed locations for multiband antennas. The desired modal weighting coefficients αn
are derived from known performance specifications, such as the desired pattern or
even potentially the desired input admittance. Thus, the procedure can transform
real design criteria, such as the desired pattern over some frequency bandwidth, to
the characteristic mode domain, where the number and location of feed points, along
with their associated voltages, may be provided. Such a procedure was shown to be
extremely general, even suggesting geometrical modifications in the case of a folded
spherical helix antenna to enhance bandwidth.
6.1.4 Mutual Coupling Reduction
In Chapter 5, two new modal systems were introduced for the purpose of analyzing
and mitigating mutual coupling between a single antenna and one or more neighbors.
The first, called Subsystem Classical CM (SCCM), was a generalization of a single
isolated antenna’s classical characteristic modes to the situation of the CM of an
embedded antenna (i.e. an antenna operating the presence of one or more neighboring
144
antennas or structures). Using SCCM, several antennas were analyzed to demonstrate
its generality and utility.
The second modal system is called Target Coupling CM (TCCM), and it analyzes
the coupling between a single antenna (source antenna) and one or more neighboring
structures (target structure). Both SCCM and TCCM are defined to produce or-
thogonal eigencurrents over the source antenna. Based on Classical CM, only SCCM
defines orthogonal eigenpatterns, provided that all antennas/structures involved are
lossless, including any port terminations.
A method to map between the two modal systems was also introduced, which
enables an analyst to not only perform a modal analysis in either modal system, but
also informs the designer on how well certain coupling modes radiate (i.e. mapping
TCCM to SCCM). The twin modal systems were applied to the analysis of several
example antennas to verify that they operate as desired.
Finally, the modal systems were applied to first understand and then reduce the
mutual coupling between two closely spaced parallel dipoles. Three designs were de-
veloped with different tradeoffs and compared against the performance of a solution
proposed in the literature. The latter two designs, one of which is planar, provided su-
perior performance to that of the reference solution from the literature. Both TCCM
and SCCM have promise to not only analyze existing general multi-antenna systems,
but also to potentially systematically trade off mutual coupling against pattern and
impedance performance, all in a general framework exposing the underlying physics
of the problem.
145
6.2 Suggestions for Future Work
The topics discussed in this dissertation have considerable potential for further de-
velopment, especially the concepts of the feed design procedure and the systematic
analysis and design of minimal/maximal coupled antenna systems.
6.2.1 Software
Although considerable software has been written, further improvements could be
made. The most straightforward enhancement would be to develop HFSS, CST and
COMSOL simulation engine plugins to extend the compatibility and flexibility of the
UCM system. Naturally, all three proposed engines would only support network char-
acteristic mode analysis, since they none support a compatible MoM formulation.1
The next most useful enhancement would be to develop a fast, in-house Galerkin
MoM code supporting structures composed of wires, plates, and dielectric/magnetic
volumes. Such software would likely just implement the formulation discussed in a
recent dissertation [120] and use commercially available meshing software (or even
borrow the mesh from a geometry developed using a commercial electromagnetic
simulation software code such as FEKO).
Another enhancement would be design a graphical user interface to the entire
UCM system. Such an undertaking would be time-consuming, but it would likely
improve the overall ease of use. It is recommended that researchers take cues from
interfaces such as Paraview [121], or even developing a graphical language such as
Labview [122], since the UCM system supports a considerable degree of flexibility
owing to its origin as a research system, rather than a simplified commercial code.
Finally, the most significant change to the UCM system would be to port it to
1Actually, CST does have a separate MoM engine available for licensing, but current versions donot allow the export of the MoM generalized impedance matrix.
146
a platform-agnostic high-performance language, such as C++. Such a change would
also naturally enable threads or other naturally parallel computational structures to
accelerate computations. Data storage (i.e. replacing the FlatFileDatabase code)
would likely be handled well by the portable and open-source SQLite [123]. The re-
searcher would likely need to have considerable mastery of several topics in computer
science, making this last alteration less likely for the pure electromagnetic engineering
researcher.
6.2.2 Wideband Mode Tracking
In the area of wideband mode tracking, the proposed tracker has a fairly low com-
putational cost, but it assumes that all the eigenvalues and eigenvectors have been
previously computed. Unfortunately, the practical cost of performing a generalized
eigenvalue/eigenvector decomposition of a matrix pair is quite high and repeated
evaluations do not scale well for large systems involving several thousand unknowns.
While GPU acceleration has shown some promise to alleviate the temporal cost of
repeated eigenvalue decompositions, a better theoretical approach could be possible.
For future work on the tracking procedure, it is suggested that an underlying state
space model be created for each eigenvector and its coefficients computed. Based on
the single parameter of frequency, the model would essentially compute a particular
mode’s eigenvector at very low cost at arbitrary frequencies. Thus, tracking would
be inherent to the model.
To build the state space model, eigenvectors would need to be explicitly computed
at several frequencies. Such frequencies would form only a small subset of the total
number of frequency points. The technology would behave very much like an adaptive
frequency sweep in popular commercial frequency-domain electromagnetic simulation
147
packages. In this way, the researcher could take advantage of the latest developments
in model order reduction techniques [124], which lie at the heart of such techniques.
6.2.3 Antenna Feed Design
While a few examples of the antenna feed design procedure were shown for some
antennas Prof. Rojas’ research group has investigated in the past using known de-
sign criteria, it would be significant if designs could be developed where the de-
sired impedance behavior and pattern bandwidth were explicitly mapped onto modal
weighting coefficients and fed into the algorithm. An outstanding problem charac-
teristic of antennas using multiple feeds requiring further research is how to match
at multiple ports over some bandwidth. At the present, the author is not aware of
any such general solutions, although there are some recent interesting developments
[125].
An unexplored area requiring some development is the application of this proce-
dure to designing MIMO antennas. In particular, it should be possible to develop a
single structure with multiple feed points. For simplicity of discussion, let each feed
point excite a unique set of classical characteristic modes. Because of orthogonality,
the feed points would be highly isolated from each other. The antenna feed design
procedure could be applied to compute each port’s location on the structure given
the information that each feed port should excite a different set of modes.
Finally, an important line of research would investigate the definition of ports on
non-wire geometries. Currently, the procedure can handle such geometries and locate
a physical port on the geometry, but these ”ports” are simply thin gaps between mesh
triangle edges. In general, an actual feed port cannot be physically constructed in such
a way, as it would usually be shorted by neighboring triangular patches. Currently,
the author uses the port locations computed by the procedure on such structures
148
as suggested places to define ports. The geometry is then modified, introducing
sufficient gaps to define a physical port. Refinements are usually necessary to excite
the modes according to the desired modal weighting coefficients. A more rigorous
approach would clarify and accelerate this process.
6.2.4 Mutual Coupling Reduction
The concepts explored by the computation of the TCCM and SCCM for a given
multiple antenna system show great promise, especially in analyzing multiple antenna
systems. Still, while the TCCM and SCCM were successfully applied to reduce the
mutual coupling between two parallel dipoles using a non-obvious geometry, the most
obvious future work is to apply them to reduce the mutual coupling among more
diverse multiantenna system geometries and arrangements.
While the antenna feed design procedure was originally developed to excite min-
imum coupling modes on an arbitrary source antenna, it was found that geometry
modification was substantially more appropriate for the design example in Chapter
5. An useful extension of the TCCM/SCCM line of research would be to develop
source antenna geometries which feature minimum coupling modes which have low
associated SCCM eigenvalue magnitudes. With lower eigenvalue magnitudes, the an-
tenna feed design procedure could be applied to develop multiport source antennas
with minimum mutual coupling to a target antenna.
While the two modal systems were used to minimize the mutual coupling between
two antennas, it is conceivable that they could also be used for antenna placement on
larger fixed structures. In particular, the TCCM could potentially be used to modify
a source antenna to have minimum induced current on some neighboring structures,
like a ship’s hull or a cellphone enclosure. Such development would be a useful
149
extension of Newman’s earlier work on antenna port placement using characteristic
modes [32].
Finally, an interesting application of the TCCM would be to develop designs to
maximize the coupling between two antennas. Such designs would allow for a general
treatment of wireless charging technology. In this case, one would seek out modes
which would have large TCCM eigenvalues and which would have minimum radiation
(i.e. larger SCCM eigenvalues). Of course, to extract a useful high transducer power
gain, the mismatch problem would need to be solved on both the source and target
antennas, implying the use of potentially small SCCM eigenvalues on both antennas.
An exploration of this tradeoff would be another useful development of these modal
systems.
150
Appendix A
ALTERNATIVE DERIVATION FOR CLASSICAL
CHARACTERISTIC MODES
This appendix presents an alternative derivation for classical characteristic modes
to [22], which begins with the requirement that modal patterns be orthogonal and
results in the defining classical characteristic mode generalized eigenvalue problem.
A.1 Derivation
Let us consider the following generalized eigenvalue problem:
[Z]Jn = γn[M ]Jn
where [Z] = [R]+j[X], [M ] are complex symmetric matrices, and γn = αn+jβn, Jn
is a generalized eigenpair. Then,
⟨Jm, [Z]Jn
⟩=⟨Jm, γn[M ]Jn
⟩⟨Jm, [R]Jn
⟩+ j
⟨Jm, [X]Jn
⟩= αn
⟨Jm, [M ]Jn
⟩+ jβn
⟨Jm, [M ]Jn
⟩If we restrict [M ] to be real,
⟨Jm, [M ]Jn
⟩=
1
αn
⟨Jm, [R]Jn
⟩⟨Jm, [M ]Jn
⟩=
1
βn
⟨Jm, [X]Jn
⟩151
But what is⟨Jm, [R]Jn
⟩or⟨Jm, [X]Jn
⟩? These are powers.
Let ( ~En, ~Hn) be the associated modal fields produced by some impressed source
~Jn. A critical property from MoM is that ~Jn is represented by the basis weighting
coefficient vector Jn and the operator Z is represented by the generalized impedance
matrix [Z] such that the following is exactly satisfied:
⟨Jm, [Z]Jn
⟩=
y
V ′
~J∗m · Z( ~Jn)dS
We want these fields to be orthogonal in the far-field. Jn exists in some closed volume
V ′, which is bounded by some surface S ′.
From Maxwell’s equations,
∇× ~En = −jωµ ~Hn (A.1.1)
∇× ~Hn = jωε ~En + ~Jn (A.1.2)
We can derive the complex power from Equations A.1.1 and A.1.2 in a standard way
[126]:
~Em · ∇ × ~H∗n − ~H∗n · ∇ × ~Em = −jωε ~Em · ~E∗n + jωµ ~H∗n · ~Hm + ~Em · ~J∗n
But
~Em · ∇ × ~H∗n − ~H∗n · ∇ × ~Em = −∇ ·(~Em × ~H∗n
)Thus,
−y
V
∇ ·(~Em × ~H∗n
)dV ′ = jω
y
V
(µ ~Hm · ~H∗n − ε ~Em · ~E∗n
)dV ′ +
y
V
~Em · ~J∗ndV ′
(A.1.3)
where V is a volume enclosing V ′, S is the closed surface of V . Let
Psource = −y
V
~Em · ~J∗ndV ′
152
Substituting A.1.3 into the above equation, we obtain:
Psource =
S
(~Em × ~H∗m
)· d~S ′ + jω
y
V
(µ ~Hm · ~H∗n − ε ~Em · ~E∗n
)dV ′ (A.1.4)
In the network domain,
Psource =⟨Jn, [Z]Jm
⟩=⟨Jn, [R]Jm
⟩+ j
⟨Jn, [X]Jm
⟩Adding Eq. A.1.4 with its conjugate of the case where n and m are interchanged, we
obtain
S
(~Em × ~H∗n
)· d~S ′ + jω
y
V
(µ ~Hm · ~H∗n − ε ~Em · ~E∗n
)dV ′
+
S
(~E∗n × ~Hm
)· d~S ′ − jω
y
V
(µ ~H∗n · ~Hm − ε ~E∗n · ~Em
)dV ′ =
⟨Jn, [R]Jm
⟩+ j
⟨Jn, [X]Jm
⟩+⟨J∗m, [R]J∗n
⟩− j
⟨J∗m, [X]J∗n
⟩Since
⟨b1, [A]b2
⟩=⟨b∗2, [A]b∗1
⟩for Hermitian [A], the R.H.S. of the above equation is
equal to
R.H.S. =⟨Jn, [R]Jm
⟩+ j
⟨Jn, [X]Jm
⟩+⟨J∗m, [R]J∗n
⟩− j
⟨J∗m, [X]J∗n
⟩= 2
⟨Jn, [R]Jm
⟩Thus,
S
(~Em × ~H∗n + ~E∗n × ~Hm
)· d~S ′ = 2
⟨Jn, [R]Jm
⟩(A.1.5)
In the far field (i.e. S → Σ, V → V∞),
~En = η ~Hn × r
~Hn =1
ηr × ~En
153
Under these conditions,
~Em × ~H∗n =1
η
(~Em × r × ~E∗n
)=
1
ηr(~Em · ~E∗n
)− ~E∗n
(~Em · r
)=
1
ηr(~Em · ~E∗n
)Thus, Eq. A.1.5 simplifies in the far-field to the following:
1
η
Σ
(~Em · ~E∗n + ~Em · ~E∗n
)r · rdS ′ = 2
⟨Jn, [R]Jm
⟩1
η
Σ
~Em · ~E∗ndS ′ =⟨Jn, [R]Jm
⟩We want ~En and ~Em to be orthogonal over the entire sphere in the far-field Σ.
Specifically,
1
η
Σ
~Em · ~E∗ndS ′ = δmn
It follows that there is a similar orthogonality between the modal magnetic fields:
η
Σ
~Hm · ~H∗ndS ′ = δmn
Therefore, the property of far-field orthogonality requires⟨Jn, [R]Jm
⟩= δmn
This condition implies ⟨Jm, [M ]Jn
⟩=
1
αnδmn
If we let αn = 1 and βn = λn, then⟨Jm, [M ]Jn
⟩=⟨Jm, [R]Jn
⟩The above relationship is obviously satisfied if we let [M ] = [R]. Then,
[Z]Jn = ([R] + j[X])Jn
= (1 + jλn)[R]Jn
154
Simplifying, we finally obtain the defining generalized eigenvalue problem for classical
characteristic modes:
[X]Jn = λn[R]Jn
Notice that the above GEP implies far-field orthogonality in addition to orthog-
onality over the source domain (by virtue of the spectral theorem [93, pg. 296]).
155
Appendix B
MODAL INPUT POWER IN CLASSICAL
CHARACTERISTIC MODES
This appendix discusses the modal decomposition of total input power in classical
characteristic mode analysis.
B.1 Derivation
In general, the input power for an antenna in MoM can be described by [23]
Pin =1
2Re⟨J , [Z]J
⟩To decompose the input power in terms of classical characteristic modes, we begin
by decomposing the total current into a weighted sum of modal currents:
J =N∑n
αnJn
By linearity, it follows that
[Z]J =N∑n
αn(1 + jλn)[R]Jn
Computing the complex total power:
⟨J , [Z]J
⟩=
N∑n
α∗m
⟨Jm,
N∑n
αn(1 + jλn)[R]Jn
⟩
=N∑n
α∗m
N∑n
αn(1 + jλn)⟨Jm, [R]Jn
⟩156
Since Jn are [R] orthogonal, the complex total power simplifies to:
⟨J , [Z]J
⟩=
N∑n
|αm|2 (1 + jλm)⟨Jm, [R]Jm
⟩Therefore, the total input power for the antenna is given by
Pin =1
2
N∑n
|αn|2⟨Jn, [R]Jn
⟩If we assume that the modal currents have been normalized such that
⟨Jn, [R]Jn
⟩,
then the total input power expression simplifies further:
Pin =1
2
N∑n
|αn|2 (B.1.1)
157
Appendix C
MODAL WEIGHTING COEFFICIENT DERIVATION
FOR GENERAL MODAL SYSTEMS
This appendix discusses the modal weighting coefficient definition for general modal
systems. We begin by discussing an alternate formula for the modal weighting coef-
ficient in classical characteristic modes.
C.1 General Modal Systems
In all characteristic mode related modal systems, we can decompose a total current
into a weighted sum of modal vectors (usually modal currents, so we shall denote
them by Jn):
J =N∑n
αnJn
But what are the αn’s and how are they related to the excitation field vector Ei?
From MoM, recall that
Ei = [Z]J
158
C.2 Classical CM
Classical characteristic modes are defined by the following generalized eigenvalue
problem
[X]Jn = λn[R]Jn (C.2.1)
This generalized eigenvalue problem may be equivalently stated as (see A)
[Z]Jn = ([R] + j[X])Jn = (1 + jλn)[R]Jn
Then, ⟨Jn, E
i⟩
=⟨Jn, [Z]J
⟩=
N∑m
αm⟨Jn, [Z]Jm
⟩=
N∑m
αm(1 + jλm)⟨Jn, [R]Jm
⟩= αn(1 + jλn)
The last step uses the fact that Jn are R orthogonal in classical characteristic modes
and that we assume that Jn has been normalized such that⟨Jn, [R]Jn
⟩= 1. Finally,
we can solve for the modal weighting coefficient αn:
αn =
⟨Jn, E
i⟩
1 + jλn(C.2.2)
The above equation is the standard modal weighting coefficient formula for classical
characteristic modes found in the literature [22]. If we instead assume that the MoM
problem is defined as
Ei = −[Z]J
or equivalently1, [L( ~J) + ~Ei
]tan
= 0,
1We have been writing ~Ei instead of ~Eitan for convenience
159
then the modal weighting coefficient formula is instead:
αn = −⟨Jn, E
i⟩
1 + jλn(C.2.3)
Modes from ESP5 [95] use Eq. C.2.2, while modes from FEKO should use Eq. C.2.3.
To ensure equivalence, the processing code for FEKO MoM Z matrices includes a
negative sign so that modes computed using FEKO in the UCM system also use
C.2.2.
C.2.1 Alternative Definition
There is an alternative way to compute the αn’s in classical characteristic modes. We
begin with a slightly different inner product involving only eigencurrents Jn and the
source current J : ⟨Jn, [R]J
⟩=
N∑m
αm⟨Jn, [R]Jm
⟩=
N∑m
αmδmn⟨Jn, [R]Jn
⟩= αn
⟨Jn, [R]Jn
⟩Thus, the modal weighting coefficient in classical characteristic modes is also:
αn =
⟨Jn, [R]J
⟩⟨Jn, [R]Jn
⟩ (C.2.4)
Since we normally choose the normalize the Jn such that⟨Jn, [R]Jn
⟩= 1, then
αn =⟨Jn, [R]J
⟩This way of deriving the expression of the modal weighting coefficients αn in the
source domain is important, since it is more easily related to the generalized eigenvalue
problem C.2.1. Since both the [X] and [R] matrices are real symmetric matrices (and
more generally, they are complex Hermitian matrices), the eigenvalues λn are real
and the eigenvectors Jn are orthogonal with respect to [R] and [X].
160
C.3 Generalized CM
Generalized characteristic modes are defined by the following generalized eigenvalue
problem:
[X]Jn = λn[H]Jn (C.3.1)
where
[H] = [G]H [G]
W (r, θ, φ)~F (r, θ, φ) = [G]J
and [G] is the matrix operator which maps the meshed current J onto the electric
field ~F (r, θ, φ) weighted by some real weighting function W (r, θ, φ).
Since both [X] and [H] are Hermitian ([X] is a real symmetric matrix, while [H]
is a complex Hermitian matrix) and [H] is positive definite, the eigenvalues λn are
real and the eigenvectors Jn are orthogonal with respect to [H] or [X]. Specifically,
⟨Jm, [H]Jn
⟩= 0⟨
Jm, [X]Jn⟩
= 0
for m 6= n.
To compute the modal weighting coefficients αn for generalized characteristic
modes, we can use these orthogonality relationships. For [M ] = [H] or [M ] = [X],
we have
⟨Jn, [M ]J
⟩=
N∑n
αm⟨Jm, [M ]Jn
⟩=
N∑m
αmδmn⟨Jm, [M ]Jn
⟩= αn
⟨Jn, [M ]Jn
⟩161
Thus, the modal weighting coefficients αn for generalized characteristic modes are
given by
αn =
⟨Jn, [M ]J
⟩⟨Jn, [M ]Jn
⟩ (C.3.2)
where [M ] = [H] or [M ] = [X].
C.4 Other CM Systems
Other compatible characteristic mode analysis systems may be specified as
[N ]Jn = λn[D]Jn (C.4.1)
Assuming that [N ] and [D] are complex Hermitian matrices and [D] is a positive
definite matrix, the eigenvectors Jn are orthogonal with respect to [N ] and [D].
Generalizing from the previous section, if we let [M ] = [N ] or [M ] = [D], we may
similarly derive the modal weighting coefficients αn as
αn =
⟨Jn, [M ]J
⟩⟨Jn, [M ]Jn
⟩ (C.4.2)
Notice that [M] is not necessarily related to the impedance matrix [Z], which
implies that the modal weighting coefficients are not directly related to Ei. We can
explicitly show the modal weighting coefficient dependence on Ei (assuming that Jn
is a modal current) by noting that
J = [Y ]Ei
Thus, the modal weighting coefficients are restated as
αn =
⟨Jn, [M ][Y ]Ei
⟩⟨Jn, [M ]Jn
⟩ (C.4.3)
Equation C.4.3 is the most general expression for the modal weighting coefficient that
explicitly depends upon the excitation vector Ei.
162
Appendix D
MODAL WEIGHTING COEFFICIENT DERIVATION
USING MODAL PATTERNS
For modal systems defining far-field modal orthogonality such as CCM, it is possible
to decompose a total field into a weighted superposition of modal fields:
~E =N∑n
αn ~En
The computation of the weights αn can be achieved due to the orthogonality of the
modal fields ⟨~Em, ~En
⟩Σ
= 0 for m 6= n
where the inner product is defined as
⟨~Em, ~En
⟩Σ≡
2π∫0
dφ
π∫0
~E∗m ~En sin θdθ (D.0.1)
The far-field orthogonality is an essential feature of especially CCM theory, enabling
us to derive the value of the weights αn. A brief derivation of the modal weights is
provided below and is followed by important numerical considerations.
D.1 Derivation of Modal Weights
Let some total far-field pattern ~E be a weighted superposition of modal fields~En
:
~E =N∑n
αn ~En
163
Now, compute the inner product of this total pattern with some particular modal
far-field pattern ~Em:
⟨~Em, ~E
⟩Σ
=
⟨~Em,
N∑n
αn ~En
⟩Σ
=N∑n
αn
⟨~Em, ~En
⟩Σ
= αm
⟨~Em, ~Em
⟩Σ
since⟨~Em, ~En
⟩Σ
= 0 for m 6= n.
Thus, the modal weight is simply
αn =
⟨~En, ~E
⟩Σ⟨
~En, ~En
⟩Σ
(D.1.1)
D.2 Numerical Considerations
To reliably use the Equation D.1.1 to compute the modal weights using the modal
patterns, it is required that the modal pattern orthogonality holds for all N modes. If
it does not not (due to numerical errors), then Equation D.1.1 is at best approximate.
The formula requires the precise computation of two inner products, both in-
volving Equation D.0.1. The denominator of Equation D.1.1 is concerned with the
radiated power of the modal pattern. The error from the double integral underlying
this inner product (Equation D.0.1), approximated using the trapezoidal rule, de-
pends only on the magnitude of ~En, rarther than the magnitude and phase. Thus,
the denominator is fairly stable in practice, since the modal pattern magnitude is
typically much less susceptible to numerical noise.
The numerator of Equation D.1.1, however, involves a double integral which does
depend on both the magnitude and phase of the two modal patterns. In order for
modal pattern orthogonality to be satisfied, the phases of ~En and ~E must be known
164
accurately, such that the outcome of the integral represents only the projection of ~E
onto ~En. If the pattern phases have some small error, then the result of the inte-
gral shall involve some contributions from the other modal patterns. Unfortunately,
the actual pattern phases obtained from several numerical EM solvers lack sufficient
precision, causing modal orthogonality to degrade, especially for the high-directivity
modal patterns radiated by higher order modes.
How do we practically address this serious numerical problem? We return to the
above derivation, but assume that the patterns are not orthogonal.
⟨~Em, ~E
⟩Σ
=N∑n
αn
⟨~Em, ~En
⟩Σ
The above equation actually represents N linearly independent equations (m = 1 to
N). Let
αm =
⟨~Em, ~E
⟩Σ⟨
~Em, ~Em
⟩Σ
Then, we can represent the N linear equations in matrix form:
¯α = [C]α
where
Cmn =
⟨~Em, ~En
⟩Σ⟨
~Em, ~Em
⟩Σ
α =
α1
...
α2
¯α =
α1
...
α2
Ideally, [C] is the identity matrix (for orthogonal modal patterns), but any spillover
165
in the projection of one modal pattern on another is also recorded using this formula-
tion. Therefore, to obtain the actual weighting coefficients αn from the raw pattern
weight coefficients αn, we can simply invert the matrix [C]:
α = [C]−1 ¯α
The matrix [C] is referred to as the eigenpattern calibration matrix. An important
property to notice about [C] is that it is almost Hermitian:
Cmn =
⟨~Em, ~En
⟩Σ⟨
~Em, ~Em
⟩Σ
Cnm =
⟨~En, ~Em
⟩Σ⟨
~En, ~En
⟩Σ
=
⟨~Em, ~En
⟩∗Σ⟨
~En, ~En
⟩Σ
The only case in which Cmn = C∗nm is when⟨~Em, ~Em
⟩Σ
=⟨~En, ~En
⟩Σ
. That is, if
each modal field is normalized such that⟨~Em, ~Em
⟩Σ
= K (for some real positive K)
for any m, then [C] is a Hermitian matrix. This property can save considerable time
in computing [C] at each frequency.
166
Appendix E
MUTUAL IMPEDANCE DERIVATION
The mutual impedance of a two port system ZP21 is defined as:
ZP21 =
V oc2
I1
∣∣∣∣I1=0
The expression of mutual impedance involving electromagnetic quantities may be
derived using the Lorentz reciprocity principle [126, pp. 118-119]. The reciprocity
principle is applied to the two problems (a) and (b) shown in Figure E.1. From
reciprocity, we have ∫VAnt. 2
~EA2 · ~JB2 dV =
∫VAnt. 2
~EB2 · ~JA2 dV
Note that ∫VAnt. 2
~EA2 · ~JB2 dV =
∫Port 2
~EA2 · IB2 d~l = −V A
2 IB2
which implies that the open-circuit voltage at port 2 in Figure E.1(a) is identical to
V oc2 :
V oc2 = V A
2 = − 1
IB2
∫VAnt. 2
~EA2 · ~JB2 dV
From the definition of mutual impedance, we have
Z21 =V oc
2
I1
∣∣∣∣I2=0
=V A
2
I1
= − 1
I1IB2
∫VAnt. 2
~EA2 · ~JB2 dV
If we let the terminal currents be the same Ip = I1 = IB2 , then we arrive at the final
expression for mutual impedance:
Z21 = − 1
(Ip)2
∫VAnt. 2
~EA2 · ~JB2 dV (E.0.1)
167
IA1!EA1
!JA1
!JA2
!EA2
(a)
IB2!JB2
!EB2
!EB1
!JB1
(b)
Ant. 1
Ant. 2
Figure E.1: Two problems for mutual impedance
where ~EA2 is the total induced electric field across port 2 due to the impressed current
density ~JA1 on antenna 1, generated by some impressed current source I1 at port 1,
provided that port 2 is open-circuited; and ~JB2 is the impressed current density on
antenna 2, generated by an impressed current source IB2 at port 2, provided that port
1 is open-circuited.
This expression E.0.1 is the same as Equation (24) from [84].
168
BIBLIOGRAPHY
[1] S. Best, “The radiation properties of electrically small folded spherical helixantennas,” IEEE Trans. Antennas Propagat., vol. 52, no. 4, pp. 953 – 960,Apr. 2004.
[2] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Wiley-Interscience,April 2005.
[3] V. Rumsey, “Frequency independent antennas,” in IRE International Conven-tion Record, vol. 5. IEEE, 1966, pp. 114–118.
[4] J. Dyson, “The equiangular spiral antenna,” IRE Transactions on Ant. andProp., vol. 7, no. 2, pp. 181–187, 1959.
[5] ——, “The characteristics and design of the conical log-spiral antenna,” IEEETrans. Antennas Propagat., vol. 13, no. 4, pp. 488–499, 1965.
[6] R. DuHamel and F. Ore, “Logarithmically periodic antenna designs,” in IREInternational Convention Record, vol. 6. IEEE, 1958, pp. 139–151.
[7] S. Best and J. Morrow, “On the significance of current vector alignment inestablishing the resonant frequency of small space-filling wire antennas,” An-tennas and Wireless Propagation Letters, IEEE, vol. 2, pp. 201–204, 2005.
[8] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Design of antenna conformal tov-shaped tail of uav based on the method of characteristic modes,” in EWCABerlin, 2009.
[9] R. Mailloux, “Conformal array antenna theory and design [reviews and ab-stracts],” IEEE Antennas and Propagation Magazine, vol. 49, no. 5, pp. 126–127, 2007.
[10] H. Rajagopalan and Y. Rahmat-Samii, “A novel conformal all-surface mountrfid tag antenna design,” in Antennas and Propagation Society InternationalSymposium, 2009. APSURSI ’09. IEEE, 2009, pp. 1–4.
[11] R. E. Collin, Field theory of guided waves, 2nd ed. New York: IEEE Press,1991.
169
[12] D. Pozar, Microwave Engineering, 3rd ed. John Wiley & Sons, Hoboken, NJ,USA, 2005.
[13] C. E. Baum, “On the singularity expansion method for the solution of electro-magnetic interaction problems,” AFWL Interaction Notes 88, Tech. Rep., Dec.1971.
[14] J. He and Z.-F. Fu, Modal analysis. Oxford: Butterworth-Heinemann, 2001.
[15] G. Pelosi, “The finite-element method, part i: Rl courant [historical corner],”IEEE Antennas and Propagation Magazine, vol. 49, no. 2, pp. 180–182, 2007.
[16] A. K. Chopra and R. K. Goel, “A modal pushover analysis procedure for esti-mating seismic demands for buildings,” Earthquake Engineering and StructuralDynamics, vol. 31, pp. 561–582, 2002.
[17] A. K. Chopra, Dynamics of Structures, 3rd ed. Prentice Hall, September 2006.
[18] W. Ren and T. Zhao, “Experimental and analytical modal analysis of steel archbridge,” Journal of Structural Engineering, vol. 130, p. 1022, 2004.
[19] R. D’Vari and M. Bakert, “Aeroelastic loads and sensitivity analysis for struc-tural loads optimization,” Journal of Aircraft, vol. 36, no. 1, 1999.
[20] R. Garbacz, “Modal expansions for resonance scattering phenomena,” Proc.IEEE, vol. 53, pp. 856–864, Aug. 1965.
[21] R. F. Harrington, Field Computation by Moment Methods. Wiley-IEEE Press,April 1993.
[22] R. Harrington and J. Mautz, “Theory of characteristic modes for conductingbodies,” IEEE Trans. Antennas Propagat., vol. AP-19, no. 5, pp. 622–628,Sept. 1971.
[23] ——, “Computation of characteristic modes for conducting bodies,” IEEETrans. Antennas Propagat., vol. AP-19, no. 5, pp. 629–639, September 1971.
[24] R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated andscattered fields,” IEEE Trans. Antennas Propagat., vol. 19, no. 3, pp. 348–358,May 1971.
[25] R. Harrington and J. Mautz, “Control of radar scattering by reactive loading,”IEEE Trans. Antennas Propagat., vol. AP-20, no. 4, pp. 446–454, July 1972.
[26] ——, “Pattern synthesis for loaded n-port scatterers,” IEEE Trans. AntennasPropagat., vol. AP-22, no. 2, pp. 184–190, Mar. 1974.
[27] ——, “Modal analysis of loaded n-port scatterers,” IEEE Trans. AntennasPropagat., vol. AP-21, no. 2, pp. 188–199, Mar. 1973.
170
[28] R. Harrington, J. Mautz, and Y. Chang, “Characteristic modes for dielectricand magnetic bodies,” IEEE Trans. Antennas Propagat., vol. 20, no. 2, pp.194 – 198, Mar. 1972.
[29] Y. Chang and R. Harrington, “A surface formulation for characteristic modesof material bodies,” IEEE Trans. Antennas Propagat., vol. 25, no. 6, pp. 789– 795, Nov. 1977.
[30] A. H. Nalbantoglu, “New computation method for characteristic modes,” Elec-tronics Letters, vol. 18, no. 23, pp. 994–996, Nov. 1982.
[31] A. O. Yee and R. J. Garbacz, “Self- and mutual-admittances of wire antennasin terms of characteristic modes,” IEEE Trans. Antennas Propagat., vol. 21,no. 6, pp. 868–871, Nov. 1973.
[32] E. Newman, “Small antenna location synthesis using characteristic modes,”IEEE Trans. Antennas Propagat., vol. 27, no. 4, pp. 530–531, Jul. 1979.
[33] R. J. Garbacz and D. M. Pozar, “Antenna shape synthesis using characteristicmodes,” IEEE Trans. Antennas Propagat., vol. 30, no. 3, pp. 340–350, May1982.
[34] F. Harackiewicz and D. Pozar, “Optimum shape synthesis of maximum gainomnidirectional antennas,” IEEE Trans. Antennas Propagat., vol. 34, no. 2,pp. 254–258, 1986.
[35] N. Inagaki and R. Garbacz, “Eigenfunctions of composite hermitian operatorswith application to discrete and continuous radiating systems,” IEEE Trans.Antennas Propagat., vol. 30, no. 4, pp. 571–575, 1982.
[36] D. Pozar, “Antenna synthesis and optimization using weighted inagaki modes,”IEEE Trans. Antennas Propagat., vol. 32, no. 2, pp. 159 – 165, February 1984.
[37] D. Liu, “Some relationships between characteristic modes and inagaki modesfor use in scattering and radiation problems,” Master’s thesis, The Ohio StateUniversity, 1986.
[38] D. Liu, R. Garbacz, and D. Pozar, “Antenna synthesis and optimization usinggeneralized characteristic modes,” IEEE Trans. Antennas Propagat., vol. 38,no. 6, pp. 862–868, June 1990.
[39] J. Ethier and D. McNamara, “The use of generalized modes in the design ofmimo antennas,” IEEE Trans. Magnetics, vol. 45, pp. 1124–1127, 2009.
[40] R. Harrington and J. Mautz, “Characteristic modes for aperture problems,”IEEE Microwave Theory Tech., vol. 33, no. 6, pp. 500–505, Jun. 1985.
171
[41] Y. Leviatan, “Low-frequency characteristic modes for aperture coupling prob-lems,” IEEE. Trans. Microwave Theory and Tech., vol. 34, no. 11, pp. 1208–1213, November 1986.
[42] A. El-Hajj, K. Y. Kabalan, and R. F. Harrington, “Characteristic modes of aslot in a conducting cylinder and their use for peneration and scattering, tecase,” IEEE Trans. Antennas Propagat., vol. 40, no. 2, pp. 156–161, February1992.
[43] A. El-Hajj and K. Y. Kabalan, “Characteristic modes of a rectangular aperturein a perfectly conducting plane,” IEEE Trans. Antennas Propagat., vol. 42,no. 10, pp. 1447–1450, October 1994.
[44] M. Vrancken and G. A. Vandenbosch, “Characteristic aperture modes for themutual coupling analysis in finite arrays of aperture-coupled antennas,” AEU- International Journal of Electronics and Communications, vol. 56, no. 1, pp.19–26, 2002.
[45] O. M. Bucci and G. D. Massa, “Use of characteristic modes inmultiple-scattering problems,” Journal of Physics D: Applied Physics,vol. 28, no. 11, p. 2235, 1995. [Online]. Available: http://stacks.iop.org/0022-3727/28/i=11/a=003
[46] G. Amendola, G. Angiulli, and G. Di Massa, “Numerical and analytical charac-teristic modes for conducting elliptic cylinders,” Microwave and Optical Tech-nology Letters, vol. 16, no. 4, pp. 243–249, 1997.
[47] G. Angiulli, G. Amendola, and G. Massa, “Characteristic modes in multi-ple scattering by conducting cylinders of arbitrary shape,” Electromagnetics,vol. 18, no. 6, pp. 593–612, 1998.
[48] G. Angiulli and G. D. Massa, “Mutual coupling evaluation in microstrip arraysby characteristic modes,” in IEEE Antennas Propagat. Int. Symp., Davos,2000.
[49] G. Angiulli, G. Amendola, and G. D. Massa, “Application of characteristicmodes to the analysis of scattering from microstrip antennas,” Journal of Elec-tromagnetic Waves and Applications, vol. 14, pp. 1063–1081, 2000.
[50] Y. Wang, Y. Bo, G. Ji, D. Ben, G. Jiang, and W. Cao, “Hybrid technique offast rcs computation with characteristic modes and awe,” IEEE Ant. WirelessProp. Letters, vol. 6, pp. 464–467, 2007.
[51] G. Angiulli, G. Amendola, and G. Di Massa, “Application of higham-chengalgorithm to the generalised eigenproblem in computational electromagnetics,”Electronics Letters, vol. 37, no. 5, pp. 282 –283, March 2001.
172
[52] S. Cheng and N. Higham, “The nearest definite pair for the Hermitian gener-alized eigenvalue problem,” Linear Algebra and its Applications, vol. 302, pp.63–76, 1999.
[53] M. Cabedo-Fabres, A. Valero-Nogueira, J. I. Herranz-Herruzo, and M. Ferrando-Bataller, “A discussion on the characteristic mode theory limitations and itsimprovements for the effective modeling of antennas and arrays,” in IEEE Ant.and Propagat. Symposium, 2004, pp. 121–124.
[54] D. Strohschein, “Application of characteristic mode analysis to variable antennaplacement on devices operating in the near-resonant range,” Ph.D. dissertation,University of New Hampshire, 2002.
[55] D. Strohschein, K. Sivaprasad, and J. Bernhard, “Investigation of the relation-ship between the characteristic modes of isolated near-resonant sized structuresand a system of these structures,” in ITG FACHBERICHT. VDE; 1999, 2003,pp. 99–102.
[56] M. Cabedo-Fabres, A. Valero-Nogueira, and M. Ferrando-Bataller, “Systematicstudy of elliptical loop antennas using characteristic modes,” in IEEE Ant. andPropagat. Symposium, vol. 1, 2002, pp. 156 – 159.
[57] M. Cabedo-Fabres, A. Valero-Nogueira, E. Antonino-Daviu, and M. Ferrando-Bataller, “Modal analysis of a radiating slotted pcb for mobile handsets,” inThe European Conference on Antennas and Propagation: EuCAP 2006, ser.ESA Special Publication, vol. 626, 2006.
[58] M. Hilbert, M. Tilston, and K. Balmain, “Resonance phenomena of log-periodicantennas: characteristic-mode analysis,” IEEE Transactions on Antennas andPropagation, vol. 37, pp. 1224–1234, 1989.
[59] J. Ethier and D. McNamara, “Modal significance measure in characteristicmode analysis of radiating structures,” Electronics Letters, vol. 46, no. 2, Jan.2010.
[60] ——, “Multiband antenna synthesis using characteristic mode indicators as anobjective function for optimization,” in IEEE Int. Conf. Wireless Info. Tech.and Systems (ICWITS). IEEE, 2010, pp. 1–4.
[61] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Discussion of series and parallelresonance phenomena in the input impedance of antennas,” Radio Science,vol. 45, December 2010.
[62] K. P. Murray and B. A. Austin, “Hf antennas on vehicles-a characteristic modalapproach,” in IEE Colloquium on HF Antennas Systems. IET, Dec. 1992, pp.2–4.
173
[63] B. A. Austin and K. P. Murray, “Modelling and design of vehicle nvis antennasystems using characteristic modes,” in Int. Conf. HF Radio Systems andTechniques. IET, 1994, pp. 207–211.
[64] K. Murray, “The design of antenna systems on complex structures using char-acteristic modes,” Ph.D. dissertation, University of Liverpool, 1993.
[65] B. A. Austin and K. P. Murray, “The application of characteristic-mode tech-niques to vehicle-mounted nvis antennas,” IEEE Antennas and PropagationMagazine, vol. 40, no. 1, pp. 7–21, Feb. 1998.
[66] E. Antonino-Daviu, M. Cabedo-Fabres, M. Ferrando-Bataller, and A. Valero-Nogueira, “Wideband double-fed planar monopole antennas,” Electronics Let-ters, vol. 39, pp. 1635–1636, 2003.
[67] M. Ferrando-Bataller, M. Cabedo-Fabres, E. Antonino-Daviu, and A. Valero-Nogueira, “Overview of planar monopole antennas for uwb applications,” inProceedings of the European Conference on Antennas and Propagation (Eu-CAP), 2006.
[68] E. Antonino-Daviu, C. A. Suarez-Fajardo, M. Cabedo-Fabres, and M. Ferrando-Bataller, “Wideband antenna for mobile terminals based on the handset pcbresonance,” Microwave Opt. Technol. Letters, vol. 48, no. 7, pp. 1408–1411,Jul. 2006.
[69] M. Ferrando-Bataller, E. Antonino-Daviu, M. Cabedo-Fabres, and A. Valero-Nogueira, “Uwb antenna design based on modal analysis,” in EuCAP. IEEE,2009, pp. 3530–3534.
[70] E. Antonino-Daviu, M. Fabres, M. Ferrando-Bataller, and V. Penarrocha, “Modalanalysis and design of band-notched uwb planar monopole antennas,” IEEETrans. Antennas Propagat., vol. 58, no. 5, pp. 1457–1467, 2010.
[71] M. Cabedo-Fabres, E. Antonino-Daviu, A. Valero-Nogueira, and M. Ferrando-Bataller, “The theory of characteristic modes revisited: A contribution to designof antennas for modern antennas,” IEEE Antennas and Propagation Magazine,vol. 49, no. 5, pp. 52–68, Oct. 2007.
[72] K. Obeidat, B. D. Raines, and R. G. Rojas, “Antenna design and analysisusing characteristic modes,” in Antennas and Propagation Society InternationalSymposium, 2007, pp. 5993–5996.
[73] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Broadband antenna synthesisusing characteristic modes,” in URSI North American Radio Science Meeting,2007.
174
[74] ——, “Application of characteristic modes and non-foster multiport loadingto the design of broadband antennas,” IEEE Trans. Antennas Propagat., vol.AP-58, no. 1, pp. 203–207, January 2010.
[75] K. Obeidat, B. D. Raines, B. T. Strojny, and R. G. Rojas, “Design of frequencyreconfigurable antennas using the theory of characteristic modes,” IEEE Trans.Antennas Propagat., vol. AP-58, no. 10, pp. 3106–3113, October 2010.
[76] B. D. Raines and R. G. Rojas, “Design of multiband reconfigurable antennas,”in EuCAP, April 2010.
[77] B. D. Raines, K. A. Obeidat, and R. G. Rojas, “Characteristic mode-baseddesign and analysis of an electrically small planar spiral antenna with omnidi-rectional pattern,” in IEEE Ant. and Propagat. Symposium, July 2008.
[78] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Design and analysis of a helicalspherical antenna using the theory of characteristic modes,” in IEEE Ant. andPropagat. Symposium, July 2008.
[79] ——, “Design of omnidirectional electrical small vee-shaped antenna using char-acteristic modes,” in URSI 2008 National Radio Science Meeting, January 2008.
[80] K. A. Obeidat, “Design methodology for wideband electrically small antennasbased on the theory of characteristic modes (cm),” Ph.D. dissertation, TheOhio State University, 2010.
[81] J. Ethier, E. Lanoue, and D. McNamara, “Mimo handheld antenna design ap-proach using characteristic mode concepts,” Microwave Opt. Technol. Letters,vol. 50, no. 7, pp. 1724–1727, July 2008.
[82] A. Ludwig, “Mutual coupling, gain and directivity of an array of two identicalantennas,” IEEE Trans. Antennas Propagat., vol. 24, no. 6, pp. 837–841, Nov.1976.
[83] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstripantennas,” IEEE Trans. Antennas Propagat., vol. 30, no. 6, pp. 1191–1196,Nov. 1982.
[84] V. B. Erturk and R. G. Rojas, “Efficient analysis of input impedance andmutual coupling of microstrip antennas mounted on large coated cylinders,”IEEE Trans. Antennas Propagat., vol. 51, no. 4, pp. 739–749, April 2003.
[85] H. Hui, “A practical approach to compensate for the mutual coupling effect inan adaptive dipole array,” Antennas and Propagation, IEEE Transactions on,vol. 52, no. 5, pp. 1262–1269, May 2004.
175
[86] B. H. Wang and H. T. Gui, “Wideband mutual coupling compensation forreceiving antenna arrays using the system identification method,” IET Mi-crowaves, Ant. and Prop., vol. 5, no. 2, pp. 184–191, 2011.
[87] Y. Rikuta, H. Arai, and Y. Ebine, “Mutual coupling suppression of two dipoleantennas backed by optimized reflector,” in IEEE Antennas Propagat. Int.Symp., vol. 2. IEEE, 2002, pp. 276–279.
[88] K. Min, D. Kim, and Y. Moon, “Improved mimo antenna by mutual couplingsuppression between elements,” in The European Conference on Wireless Tech-nology. IEEE, 2005, pp. 125–128.
[89] C. Chiu, C. Cheng, R. Murch, and C. Rowell, “Reduction of mutual couplingbetween closely-packed antenna elements,” IEEE Trans. Antennas Propagat.,vol. 55, no. 6, pp. 1732–1738, Jun. 2007.
[90] A. Mak, C. Rowell, and R. Murch, “Isolation enhancement between two closelypacked antennas,” IEEE Trans. Antennas Propagat., vol. 56, no. 11, pp. 3411–3419, Nov. 2008.
[91] K. Buell, H. Mosallaei, and K. Sarabandi, “Metamaterial insulator enabledsuperdirective array,” IEEE Trans. Antennas Propagat., vol. 55, no. 4, pp.1074–1085, April 2007.
[92] I. Mathworks. Matlab website. [Online]. Available: http://www.mathworks.com
[93] G. Strang, Linear Algebra and Its Applications, 4th ed. Brooks Cole, July2005.
[94] D. H. Brandwood, “A complex gradient operator and its application in adaptivearray theory,” Microwaves Optics and Antennas IEE Proceedings H, vol. 130,no. 1, pp. 11–16, 1983.
[95] E. Newman. The electromagnetic surface patch code: Version 5, The OhioState University. [Online]. Available: http://esl.eng.ohio-state.edu
[96] E. S. . Systems. Feko website. [Online]. Available: http://www.feko.info
[97] A. Technologies, “Agilent technologies homepage.” [Online]. Available:http://www.agilent.com
[98] T. Tanaka, “Fast generalized eigenvector tracking based on the power method,”IEEE Signal Processing Letters, vol. 16, no. 11, pp. 969 –972, November 2009.
[99] R. E. Kalaba, K. Spingarn, and L. Tesfatsion, “Individual tracking of an eigen-value and eigenvector of a parameterized matrix,” Nonlinear Analysis, vol. 5,no. 4, pp. 337–340, 1981.
176
[100] R. Alden and F. Qureshy, “Eigenvalue tracking due to parameter variation,”IEEE Trans. Automatic Control, vol. 30, no. 9, pp. 923 – 925, September 1985.
[101] P. Comon and G. Golub, “Tracking a few extreme singular values and vectorsin signal processing,” Proc. IEEE, vol. 78, no. 8, pp. 1327–1343, Aug. 1990.
[102] E. Newman, “Generation of wide-band data from the method of moments byinterpolating the impedance matrix,” IEEE Trans. Antennas Propagat., vol. 36,no. 12, pp. 1820–1824, Dec. 1988.
[103] K. Naishadham, T. W. Nuteson, and R. Mittra, “Parametric interpolation ofthe moment matrix in surface integral equation formulation,” InternationalJournal of RF and Microwave Computer-Aided Eng., vol. 9, no. 6, pp. 474–489, Oct. 1999.
[104] R. Allemang, “The modal assurance criterion – twenty years of use and abuse,”Sound and Vibration, pp. 14–21, Aug. 2003.
[105] T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms,3rd ed. The MIT Press, September 2009.
[106] E. S. . Systems, “Private corresponding with author,” Aug. 2009, fEKO MoMZ matrix symmetry.
[107] M. Bozzi, D. Li, S. Germani, L. Perregrini, and K. Wu, “Analysis of nrd com-ponents via the order-reduced volume-integral-equation method combined withthe tracking of the matrix eigenvalues,” IEEE. Trans. Microwave Theory andTech., vol. 54, no. 1, pp. 339–347, January 2006.
[108] G. Hammerlin, K.-H. Hoffmann, and L. Schumaker, Numerical Mathematics,1st ed. Springer, 1991.
[109] H. Carlin and P. Civalleri, Wideband Circuit Design. CRC, 1997.
[110] B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J.Comput., vol. 24, pp. 227–234, 1995.
[111] D. L. Donoho, “For most large underdetermined systems of linear equationsthe minimal 1-norm solution is also the sparsest solution,” Comm. Pure Appl.Math, vol. 59, pp. 797–829, 2004.
[112] Y. Zhang, “User’s guide for yall1: Your algorithms for l1 optimization,” RiceUniversity, Tech. Rep. CAAM TR09-17, 2009.
[113] J. Parker, Algorithms for image processing and computer vision, 1st ed. JohnWiley & Sons, Inc. New York, NY, USA, 1996.
177
[114] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas,”IEEE Antennas Propagat. Int. Symp., pp. 501–504, Jun. 2003.
[115] B. D. Raines, K. A. Obeidat, B. T. Strojny, and R. G. Rojas, “Design and anal-ysis of a wideband helical spherical antenna using characteristic mode theory,”in preparation, 2011.
[116] M. Jensen and J. Wallace, “A review of antennas and propagation for MIMOwireless communications,” IEEE Trans. Antennas Propagat., vol. 52, no. 11,pp. 2810–2824, 2004.
[117] M. Jones and J. Rawnick, “A new approach to broadband array design us-ing tightly coupled elements,” in Military Communications Conference, 2007.MILCOM 2007. IEEE. IEEE, 2007, pp. 1–7.
[118] D. Browne, M. Manteghi, M. Fitz, and Y. Rahmat-Samii, “Antenna topologyimpacts on measured MIMO capacity,” in IEEE Vehicular Technology Confer-ence, 2005.
[119] P. P. Viezbicke, “Yagi antenna design,” National Bureau of Standards, Techni-cal Note 688, Dec. 1976.
[120] C. Marasini, “Efficient computation techniques for Galerkin MoM antenna de-sign,” Ph.D. dissertation, Technische Universiteit Eindhoven, 2008.
[121] Kitware. Paraview. [Online]. Available: http://www.paraview.org/
[122] N. Instruments. Labview. [Online]. Available: http://www.ni.com/labview/
[123] SQLite. Sqlite homepage. [Online]. Available: http://www.sqlite.org/
[124] K. Olsson, “Model order reduction with rational Krylov Methods,” Ph.D. dis-sertation, Royal Institute of Technology, 2005.
[125] J. C. Allen, J. Rockway, and D. Arceo, “Wideband multiport matching phasei: Single-feed,” SSC San Diego, Tech. Rep. TR 1972, Sept. 2008.
[126] R. Harrington, Time-Harmonic Electromagnetic Fields. McGraw-Hill, NewYork, NY, 2001.
178